Answer: $3(x + 3)$

Expand the first product: $(x + 2)(x + 5) = x^2 + 7x + 10$. Expand the second product: $(x + 1)(x + 4) = x^2 + 5x + 4$. Subtract and add the linear term: $(x^2 + 7x + 10) - (x^2 + 5x + 4) + (x + 3) = 2x + 6 + x + 3 = 3x + 9$. Factor: $3x + 9 = 3(x + 3)$. Why distractors fail: Option A gives $2x + 6$, missing the $(x + 3)$ term. Option B gives $2x + 10$, which has the wrong constant. Option D gives $x + 9$, combining terms incorrectly.

Answer: Choose any two factors and expand their product first

Identify the key principle: Nested distribution works by reducing a multi-factor product step by step. First, pick any pair of factors and expand them into a single polynomial. Why the correct answer works: Option C correctly states the strategy: choose any two of the three factors, expand their product, then multiply the result by the third factor. Why distractors fail: Option A only computes the constant term and ignores all variable terms. Option B adds expressions that should be multiplied. Option D distributes only $x$, missing the constant terms in each binomial.

Answer: It is always valid for any values of $a$, $b$, $c$, $d$

Expand both products: $(x + a)(x + b) = x^2 + (a+b)x + ab$ and $(x + c)(x + d) = x^2 + (c+d)x + cd$. Subtract: $x^2 + (a+b)x + ab - x^2 - (c+d)x - cd = (a + b - c - d)x + (ab - cd)$. The $x^2$ terms always cancel. Why the correct answer works: Since both products have the same leading coefficient ($1 \cdot x^2$), their difference always eliminates the $x^2$ term, leaving a linear expression for any values of $a, b, c, d$. Why distractors fail: Option A adds unnecessary restrictions. Option C confuses a condition on constants with the structural cancellation. Option D is wrong because the $x^2$ terms are both $+x^2$ and always cancel when subtracted.

Answer: $x^2 + 8x - 11$

Distribute the scalar coefficients: $3(x^2 + 2x - 1) = 3x^2 + 6x - 3$ and $2(x^2 - x + 4) = 2x^2 - 2x + 8$. Subtract the second from the first: $(3x^2 + 6x - 3) - (2x^2 - 2x + 8) = 3x^2 + 6x - 3 - 2x^2 + 2x - 8 = x^2 + 8x - 11$. Why distractors fail: Option B has $+4x$ instead of $+8x$, failing to distribute the negative to $-2x$. Option C has $+5$ instead of $-11$, a sign error on the constants. Option D has $5x^2$, which results from adding instead of subtracting the $x^2$ terms.

Answer: The student did not distribute the negative sign to $-4$, giving $-4$ instead of $+4$

Compute correctly: $2(x+1)(x-3) = 2(x^2 - 2x - 3) = 2x^2 - 4x - 6$. Subtract $(x^2 - 4)$ correctly: $(2x^2 - 4x - 6) - (x^2 - 4) = 2x^2 - 4x - 6 - x^2 + 4 = x^2 - 4x - 2$. Identify the student's error: The student got $x^2 - 4x - 10$. This means they computed $-6 - 4 = -10$ instead of $-6 + 4 = -2$. They failed to distribute the negative sign to the $-4$ inside $(x^2 - 4)$, treating it as $-(x^2) - 4$ instead of $-(x^2) + 4$. Why distractors fail: Option A is wrong because $2(x+1)(x-3) = 2x^2 - 4x - 6$ is handled correctly in the student's work ($x^2$ and $x$ terms are fine). Option C is wrong because the $x^2$ coefficient of $1$ shows the factor of $2$ was applied. Option D is wrong because the correct answer is $x^2 - 4x - 2$, not $x^2 - 4x - 10$.

Answer: $2(x + 2)^2 - 2(x - 1)^2$

Verify Option D: $2(x+2)^2 = 2(x^2+4x+4) = 2x^2+8x+8$ and $2(x-1)^2 = 2(x^2-2x+1) = 2x^2-4x+2$. Subtracting: $(2x^2+8x+8)-(2x^2-4x+2) = 12x+6$. ✓ Why the other options fail: Option A: $3(x+3)^2-3(x+1)^2 = 3(x^2+6x+9)-3(x^2+2x+1) = 3(4x+8) = 12x+24$. Wrong constant. Option B: $(x+4)^2-(x+1)^2 = (x^2+8x+16)-(x^2+2x+1) = 6x+15$. Wrong coefficients entirely. Option C: $2(x+3)^2-2(x+1)^2 = 2(x^2+6x+9)-2(x^2+2x+1) = 2(4x+8) = 8x+16$. Wrong coefficients. Key insight: The general formula $a(x+p)^2 - a(x+q)^2 = a[(x+p)^2-(x+q)^2] = a(p-q)(2x+p+q)$ can speed up the check. For Option D: $a=2$, $p=2$, $q=-1$, so $2(3)(2x+1) = 6(2x+1) = 12x+6$. ✓

Answer: Both methods are valid and yield the same result, but the second is more efficient

Method 1: Expand and combine: $(2x+3)(x-1) = 2x^2 + x - 3$ and $(x+2)(x-1) = x^2 + x - 2$. Subtracting: $2x^2 + x - 3 - x^2 - x + 2 = x^2 - 1 = (x+1)(x-1)$. Method 2: Factor first: $(x-1)[(2x+3) - (x+2)] = (x-1)(2x + 3 - x - 2) = (x-1)(x + 1)$. Comparison: Both yield $(x-1)(x+1)$. Method 2 requires fewer steps and avoids full expansion. Why distractors fail: Options A and B incorrectly claim only one method works. Option D is wrong because both methods are standard algebraic techniques.

Answer: $2x^2 - 2x - 10$

Expand the first product: $(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6$. Expand the second product: $(x + 1)(x - 4) = x^2 - 4x + x - 4 = x^2 - 3x - 4$. Combine like terms: $(x^2 + x - 6) + (x^2 - 3x - 4) = 2x^2 + (1 - 3)x + (-6 - 4) = 2x^2 - 2x - 10$. Why distractors fail: Option B has $+2x$ instead of $-2x$, a sign error when combining $x$ terms. Option C has $x^2$ instead of $2x^2$, as if only one product was included. Option D has $+10$ instead of $-10$, a sign error on the constants.

Answer: $(4x + 2)(x - 3) - (3x - 1)(x - 4)$

Why a non-monic factor is necessary: The difference of two monic binomial products $(x+a)(x+b) - (x+c)(x+d)$ always cancels the $x^2$ term, giving only a linear result. To obtain a quadratic like $x^2 + 3x - 10$, at least one binomial in each product must have a leading coefficient greater than 1. Verify Option C: $(4x+2)(x-3) = 4x^2 - 10x - 6$ and $(3x-1)(x-4) = 3x^2 - 13x + 4$. Subtracting: $(4x^2 - 10x - 6) - (3x^2 - 13x + 4) = x^2 + 3x - 10$. ✓ Why the other options fail: Option A: $(4x+1)(x-3) - (3x-2)(x-4) = (4x^2-11x-3) - (3x^2-14x+8) = x^2+3x-11$. Off by 1 in the constant. Option B: $(3x+2)(x-2) - (2x-1)(x-3) = (3x^2-4x-4) - (2x^2-7x+3) = x^2+3x-7$. Wrong constant. Option D: $(4x+3)(x-3) - (3x+1)(x-4) = (4x^2-9x-9) - (3x^2-11x-4) = x^2+2x-5$. Wrong $x$ coefficient and constant.

Answer: The subtraction sign must be distributed to every term of the expression being subtracted, changing each term's sign

Identify the key principle: The computation is $(x^2 + 4x - 1) - (3x^2 - 2x + 5)$. The negative sign in front of the second group must be distributed to every term inside: $-(3x^2) -(-2x) -(5) = -3x^2 + 2x - 5$. Why the correct answer works: Option C correctly states the rule: when subtracting a grouped expression, the subtraction (negative) sign flips the sign of every term inside the parentheses. The result is $(x^2 + 4x - 1) + (-3x^2 + 2x - 5) = -2x^2 + 6x - 6$. Why distractors fail: Option A is wrong because subtracting only the leading term ignores all remaining terms. Option B is wrong because squaring terms is not part of polynomial subtraction. Option D is wrong because subtraction and addition are not interchangeable — subtracting reverses the signs of the subtracted expression, while adding does not.

Answer: $(x + 4)(x - 4)$

Factor out $(x - 4)$: All three terms share the factor $(x - 4)$: $(x - 4)[(3x + 2) - (2x - 1) + 1]$. Simplify the bracket: $(3x + 2) - (2x - 1) + 1 = 3x + 2 - 2x + 1 + 1 = x + 4$. Final result: $(x - 4)(x + 4)$. Why distractors fail: Option A gives $(x-4)(x+2) = x^2 - 2x - 8$, which does not match $x^2 - 16$. Option C is $x^2 - 16$, the expanded form, but the question asks for the factored result. Option D gives $2(x-4)(x+1) = 2x^2 - 6x - 8$, which does not equal $x^2 - 16$.

Answer: The student's answer is correct

Verify by expanding the first product: $(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$. Verify by expanding the second product: $(x - 1)(x + 4) = x^2 + 4x - x - 4 = x^2 + 3x - 4$. Combine and check: $(x^2 - x - 6) + (x^2 + 3x - 4) = 2x^2 + 2x - 10$. This matches the student's answer. Why distractors fail: Options A, C, and D all claim errors that did not occur. The student's expansion and combination are correct.

Answer: Expand and combine all like terms first, then factor the resulting simplified expression

Identify the key principle: The 'simplify then factor' strategy involves two phases: first, fully expand and collect like terms to get a standard quadratic; second, factor that simplified quadratic. Why the correct answer works: Option B accurately captures both phases: expanding and combining like terms (simplify), then factoring the result. Why distractors fail: Option A reverses the order by factoring before expanding. Option C skips the simplification step entirely. Option D confuses algebraic simplification with numerical substitution.

Answer: $x^2 + 9x - 7$

Expand the first product: $(2x - 1)(x + 3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3$. Expand the square: $(x - 2)^2 = x^2 - 4x + 4$. Subtract and simplify: $(2x^2 + 5x - 3) - (x^2 - 4x + 4) = 2x^2 + 5x - 3 - x^2 + 4x - 4 = x^2 + 9x - 7$. Why distractors fail: Option A has $+x$ instead of $+9x$, failing to distribute the negative to $-4x$. Option C has $+1$ instead of $-7$, a sign error on the constants. Option D has $3x^2$ instead of $x^2$, adding rather than subtracting the $x^2$ terms.

Answer: Multiply any two of the three factors first to get a quadratic, then multiply by the remaining factor

Identify the key principle: Nested distribution requires multiplying two factors first to produce a quadratic, then distributing the third factor across that result. Why the correct answer works: Multiplying two factors such as $(x+2)(x+3) = x^2 + 5x + 6$ gives a manageable quadratic that can then be multiplied by $(x-1)$. This is the standard nested distribution strategy. Why distractors fail: Option A is wrong because you cannot distribute a negative sign across factors that are being multiplied, not subtracted. Option C is wrong because simply adding constants does not account for the cross terms produced by distribution. Option D is wrong because the three binomials share no common factor to extract.

Answer: $12(x + 2)$

Factor out the common factor: Both terms share a factor of $4(x + 2)$: $4(x+2)^2 - 4(x+2)(x-1) = 4(x+2)[(x+2) - (x-1)]$. Simplify the bracket: $(x + 2) - (x - 1) = x + 2 - x + 1 = 3$. Final result: $4(x + 2)(3) = 12(x + 2)$. Why distractors fail: Option B gives $4(x^2 + 5x + 6) = 4x^2 + 20x + 24$, which does not equal $12x + 24$. Option C is $12x + 24$, which is the expanded form but not factored. Option D is $12x + 24$ rewritten as $4(3x + 6)$, which is not fully factored since $3$ can be factored out of $(3x + 6)$ to give $12(x + 2)$.

Answer: Both approaches yield the same final result due to the associative property of multiplication

Identify the key principle: Multiplication is associative: $(ab)c = a(bc)$. The order in which pairs of factors are multiplied does not affect the final product. Verify with Student A: $(x+1)(x+2) = x^2 + 3x + 2$, then $(x^2 + 3x + 2)(x - 3) = x^3 - 3x^2 + 3x^2 - 9x + 2x - 6 = x^3 - 7x - 6$. Verify with Student B: $(x+2)(x-3) = x^2 - x - 6$, then $(x + 1)(x^2 - x - 6) = x^3 - x^2 - 6x + x^2 - x - 6 = x^3 - 7x - 6$. Why distractors fail: Option A is wrong because the associative property guarantees the same result. Option B is wrong because there is no left-to-right requirement. Option D is wrong because both approaches are valid forms of nested distribution.

Answer: Approaches 1 and 2 are both valid algebraic methods; Approach 3 only gives one numerical value and cannot determine the full expression

Evaluate Approach 1: $(x+2)(x-3) = x^2 - x - 6$ and $(x+2)(x+1) = x^2 + 3x + 2$. Subtracting: $-4x - 8 = -4(x + 2)$. Valid. Evaluate Approach 2: $(x+2)[(x-3)-(x+1)] = (x+2)(-4) = -4(x+2)$. Valid and more efficient. Evaluate Approach 3: Substituting $x = 0$ gives $(2)(-3) - (2)(1) = -6 - 2 = -8$. This confirms the constant term at $x=0$ but cannot determine the general algebraic form of the expression. Why distractors fail: Option A is wrong because Approach 3 cannot determine the general expression. Option B is wrong because Approach 2 is also valid. Option D is wrong because substitution gives only a single point, not the full expression.

Answer: $x^3 + 3x^2 - x - 3$

Use difference of squares first: $(x + 1)(x - 1) = x^2 - 1$. Multiply by the third factor: $(x^2 - 1)(x + 3) = x^3 + 3x^2 - x - 3$. Why distractors fail: Option B has incorrect signs on every term except $x^3$. Option C has $+x$ instead of $-x$. Option D has $2x^2$ instead of $3x^2$, an arithmetic error in the distribution.

Answer: $8(x + 1)$

Expand both squares: $(x+3)^2 = x^2 + 6x + 9$ and $(x-1)^2 = x^2 - 2x + 1$. Subtract: $(x^2 + 6x + 9) - (x^2 - 2x + 1) = 8x + 8$. Factor completely: $8x + 8 = 8(x + 1)$. Why distractors fail: Option A gives $4x + 4$, which is only half the correct value — a factor of 2 is missing. Option C expands to $8x + 4$, which has an incorrect constant term. Option D expands to $8x - 8$, which has the wrong sign on the constant term.

Answer: $4(x + 2)$

Expand the first product: $(x + 5)(x + 1) = x^2 + 6x + 5$. Expand the second product: $(x + 3)(x - 1) = x^2 + 2x - 3$. Subtract and simplify: $(x^2 + 6x + 5) - (x^2 + 2x - 3) = x^2 + 6x + 5 - x^2 - 2x + 3 = 4x + 8$. Factor the result: $4x + 8 = 4(x + 2)$. Why distractors fail: Option A gives $4x + 8$ when expanded, which is equivalent, but is not fully factored since $2$ can be factored out of $(2x + 4)$. Option C gives $2x + 8$ when expanded, which is incorrect. Option D equals $8x + 8$, which does not match $4x + 8$.

Answer: It reduces a three-factor product to a standard two-factor multiplication that can be handled with known techniques

Identify the key principle: Multiplying three linear factors at once is complex. Reducing the problem by first multiplying two of them converts the task into multiplying a quadratic by a linear expression, which is a familiar process. Why the correct answer works: Option B correctly identifies that the intermediate expansion transforms a three-factor problem into a two-factor problem using standard distribution. Why distractors fail: Option A is wrong because distribution is still required in the next step. Option C is wrong because the intermediate product $x^2 + 3x + 2$ is not the final result. Option D is wrong because multiplying a quadratic by a linear factor yields a cubic, not a quadratic.

Answer: Factor out $5(x - 2)$ first, then simplify the remaining bracket

Identify the common factor: Both terms share the factors $5$ and $(x - 2)$: $5(x-2)(x+4) - 5(x+1)(x-2) = 5(x-2)[(x+4) - (x+1)]$. Simplify: $(x + 4) - (x + 1) = 3$, so the expression simplifies to $5(x-2)(3) = 15(x - 2)$. Why the correct answer works: Factoring out the common terms first avoids unnecessary expansion and leads directly to the simplified answer in fewer steps. Why distractors fail: Option A works but is less efficient, requiring many more steps. Option C is irrelevant — the quadratic formula solves equations, not simplifies expressions. Option D is unreliable and not a valid algebraic method.

Answer: $3x^2 - 2x + 3$

Combine the $x^2$ terms: $2x^2 + x^2 = 3x^2$. Combine the $x$ terms: $3x + (-5x) = -2x$. Combine the constant terms: $-4 + 7 = 3$. So the result is $3x^2 - 2x + 3$. Why distractors fail: Option A has $+2x$ instead of $-2x$, a sign error on the $x$ terms. Option C incorrectly multiplied the expressions instead of adding them. Option D has $-3$ instead of $+3$, a sign error on the constants.

Answer: The student did not distribute the negative sign to all terms in the second expression, resulting in sign errors on the $x$ and constant terms

Perform the correct subtraction: $(5x^2 + x - 3) - (2x^2 + x - 3) = 5x^2 + x - 3 - 2x^2 - x + 3 = 3x^2 + 0x + 0 = 3x^2$. Identify the student's error: The student got $3x^2 + 2x - 6$, which means they only negated the $x^2$ term and left $+x$ and $-3$ unchanged instead of flipping them to $-x$ and $+3$. Why distractors fail: Option A is wrong because the constant was addressed but with the wrong sign. Option C is wrong because the $x^2$ terms were correctly subtracted, showing subtraction was attempted. Option D is wrong because factoring before subtracting is not required.