Answer: They combined like terms by grouping $x^2$ terms and $x$ terms separately.

Identify the operation: The expression has no parentheses to expand. The student grouped $3x^2 - 2x^2 = x^2$ and $5x + x = 6x$. Why the correct answer works: Option B correctly describes combining like terms: $x^2$ terms together and $x$ terms together. Why distractors fail: Option A is wrong because there are no parentheses requiring distribution. Option C is wrong because FOIL applies to multiplying two binomials. Option D is wrong because the student added terms, not factored them.

Answer: Step 4

Check each step: Step 1: $5x \cdot x = 5x^2$ ✓. Step 2: $5x \cdot 4 = 20x$ ✓. Step 3: $-2 \cdot x = -2x$ ✓. Step 4: $-2 \cdot 4 = -8$, not $+8$. The student dropped the negative sign. Correct result: The correct expansion is $5x^2 + 20x - 2x - 8 = 5x^2 + 18x - 8$. Why distractors fail: Steps 1–3 are all correct. Step 5 follows logically from the student's (incorrect) Step 4, so the first error is in Step 4.

Answer: $(x^2 + 6x + 9)(x + 3)$

Understand the strategy: $(x + 3)^3 = (x + 3)^2 \cdot (x + 3)$. After computing $(x + 3)^2$, you multiply that result by the remaining $(x + 3)$. Why the correct answer works: Option B correctly shows the multiplication of the trinomial by the remaining binomial factor. Why distractors fail: Option A adds instead of multiplies. Option C multiplies by the scalar $3$ rather than by $(x + 3)$. Option D squares the result, giving $(x+3)^4$ instead of $(x+3)^3$.

Answer: $9x + 5$

Expand $2(x + 1)^2$: $(x + 1)^2 = x^2 + 2x + 1$, so $2(x^2 + 2x + 1) = 2x^2 + 4x + 2$. Expand $(2x + 1)(x - 3)$: $2x^2 - 6x + x - 3 = 2x^2 - 5x - 3$. Subtract and simplify: $(2x^2 + 4x + 2) - (2x^2 - 5x - 3) = 2x^2 + 4x + 2 - 2x^2 + 5x + 3 = 9x + 5$. Why distractors fail: Option B ($5x + 5$) comes from incorrectly computing $4x + x = 5x$ instead of $4x + 5x = 9x$. Option C ($4x^2 + 9x + 5$) fails to cancel the $2x^2$ terms. Option D ($-x + 5$) results from subtracting the middle terms instead of adding them when distributing the negative sign.

Answer: $(x + 5)^2$

Identify the pattern: $x^2 + 10x + 25$: check if it's a perfect square. $\sqrt{25} = 5$ and $2 \cdot x \cdot 5 = 10x$. ✓ Verify: $(x + 5)^2 = x^2 + 2(5)x + 25 = x^2 + 10x + 25$. ✓ Why distractors fail: Option A gives $x^2 - 25$ (difference of squares). Option B gives $x^2 + 50x + 625$. Option D gives $x^2 + 35x + 250$.

Answer: $4ax$

Expand each squared binomial: $(x + a)^2 = x^2 + 2ax + a^2$ and $(x - a)^2 = x^2 - 2ax + a^2$. Subtract: $(x^2 + 2ax + a^2) - (x^2 - 2ax + a^2) = x^2 + 2ax + a^2 - x^2 + 2ax - a^2 = 4ax$. Why distractors fail: Option A ($2a^2$) ignores the $x$ terms. Option C ($2x^2 + 2a^2$) incorrectly adds instead of subtracting. Option D ($0$) would be the case if the expressions were identical, but they differ in sign on the $2ax$ term.

Answer: $x^2 - 36$

Apply FOIL: F: $x^2$. O: $-6x$. I: $+6x$. L: $-36$. The middle terms cancel: $-6x + 6x = 0$. Why the correct answer works: $(x + 6)(x - 6) = x^2 - 36$. The Outer and Inner products are additive inverses. Why distractors fail: Option B has $+36$ instead of $-36$. Option C is $(x - 6)^2$. Option D incorrectly includes a $+12x$ term and a negative constant.

Answer: $-2x^2 - 4x + 6$

First expand the binomials: $(x + 3)(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3$. Distribute the $-2$: $-2(x^2 + 2x - 3) = -2x^2 - 4x + 6$. Why distractors fail: Option B has wrong signs on the last two terms. Option C gets $-6$ instead of $+6$ (forgetting that $-2 \times -3 = +6$). Option D omits the negative sign on the leading coefficient.

Answer: Student A is correct because $3(x^2 + 4x + 4) - 3x^2 - 12 = 3x^2 + 12x + 12 - 3x^2 - 12 = 12x$.

Expand $3(x + 2)^2$: $(x + 2)^2 = x^2 + 4x + 4$, so $3(x^2 + 4x + 4) = 3x^2 + 12x + 12$. Expand $3(x^2 + 4)$: $3(x^2 + 4) = 3x^2 + 12$. Subtract: $3x^2 + 12x + 12 - 3x^2 - 12 = 12x$. Student A is correct. Why distractors fail: Option A gives the right answer but a vague justification. Option B contradicts itself (states the answer is $12x$ but attributes it to Student B who said $12x - 12$). Option C correctly shows the full work and conclusion. Option D is simply incorrect.

Answer: $2x^2 + 2x$

Expand each product: $(x + 1)(x - 1) = x^2 - 1$ and $(x + 1)^2 = x^2 + 2x + 1$. Combine: $x^2 - 1 + x^2 + 2x + 1 = 2x^2 + 2x + 0 = 2x^2 + 2x$. Why distractors fail: Option B gets $2x^2 + 2$ by omitting $2x$ and miscalculating the constants. Option C omits both the $2x$ term and the constant cancellation. Option D adds the constants as $1 + 1 = 2$ instead of $-1 + 1 = 0$.

Answer: $x \cdot 3x^2 + x \cdot x + x \cdot (-4) + 2 \cdot 3x^2 + 2 \cdot x + 2 \cdot (-4)$

Apply the distributive property fully: Each term in $(x + 2)$ must multiply every term in $(3x^2 + x - 4)$. This gives $2 \times 3 = 6$ partial products. Why the correct answer works: Option A shows all six products: $x$ distributed across three terms, then $2$ distributed across three terms. Why distractors fail: Option B only multiplies some terms and ignores others. Option C incorrectly groups terms. Option D appears to have added terms without multiplying properly.

Answer: $-4$

Expand the product: $(3x + 2)(3x - 2) = 9x^2 - 4$ (difference of squares pattern). Subtract $(3x)^2$: $9x^2 - 4 - 9x^2 = -4$. Why distractors fail: Option B has the wrong sign. Option C fails to subtract $9x^2$. Option D suggests a linear result, which cannot arise from these operations.

Answer: Because squaring a binomial produces a middle term $2 \cdot x \cdot 4 = 8x$ that is missing from $x^2 + 16$.

Expand correctly: $(x + 4)^2 = (x + 4)(x + 4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16$. Why the correct answer works: Option B correctly identifies the missing middle term $8x$. The error $x^2 + 16$ comes from squaring each term individually instead of multiplying the binomial by itself. Why distractors fail: Option A is false — you absolutely can square a binomial. Option C is wrong because $4^2 = 16$ is correct for the constant. Option D confuses this with the difference of squares pattern.

Answer: $3x^2 + 17x$

Distribute each factor: $5(x^2 + 3x) = 5x^2 + 15x$ and $-2(x^2 - x) = -2x^2 + 2x$. Combine like terms: $5x^2 - 2x^2 = 3x^2$ and $15x + 2x = 17x$. Result: $3x^2 + 17x$. Why distractors fail: Option B ($3x^2 + 13x$) likely comes from subtracting $2x$ instead of adding it ($-2 \times -x = +2x$). Option C ($3x^2 + 15x$) omits the $+2x$ from the second distribution. Option D ($7x^2 + 17x$) incorrectly adds $5x^2 + 2x^2$ instead of subtracting.

Answer: $4x^2$ and $-2x^2$

Define like terms: Like terms have the same variable raised to the same exponent. Only their coefficients may differ. Why the correct answer works: $4x^2$ and $-2x^2$ both contain $x^2$, so they are like terms and can be combined: $4x^2 + (-2x^2) = 2x^2$. Why distractors fail: $3x^2$ and $5x$ have different exponents ($x^2$ vs. $x$). $7x$ and $7x^2$ also differ in exponent. $6x$ and $6y$ have different variables entirely.

Answer: Left side $= 7$, Right side $= 7$

Evaluate the left side: $(3 + 4)(3 - 2) = 7 \cdot 1 = 7$. Evaluate the right side: $3^2 + 2(3) - 8 = 9 + 6 - 8 = 7$. Why the correct answer works: Both sides equal $7$, confirming the expansion is correct. Why distractors fail: Options A, C, and D all contain at least one incorrect evaluation. If the expansion is correct, both sides must yield the same value for any input.

Answer: $p = 3$ and $q = 4$

Use the expansion pattern: $(x + p)(x + q) = x^2 + (p + q)x + pq$. So $p + q = 7$ and $pq = 12$. Find $p$ and $q$: We need two numbers that add to $7$ and multiply to $12$. $3 + 4 = 7$ and $3 \times 4 = 12$. ✓ Why distractors fail: Option A: $2 + 6 = 8 \neq 7$. Option C: $1 + 12 = 13 \neq 7$. Option D: $(-3) + (-4) = -7 \neq 7$.

Answer: $16x^2 - 8x + 1$

Apply the squared binomial formula: $(4x - 1)^2 = (4x)^2 - 2(4x)(1) + 1^2 = 16x^2 - 8x + 1$. Why the correct answer works: Option C correctly computes all three terms: $16x^2$, $-8x$, and $+1$. Why distractors fail: Option A omits the middle term $-8x$. Option B uses $-4x$ instead of $-8x$ (forgot to multiply by 2). Option D incorrectly computes $(4x)^2$ as $8x^2$ instead of $16x^2$.

Answer: $2x^2 + 8x - 12$

Distribute each factor: $x(2x + 5) = 2x^2 + 5x$ and $3(x - 4) = 3x - 12$. Combine like terms: $2x^2 + 5x + 3x - 12 = 2x^2 + 8x - 12$. Why distractors fail: Option B has $5x$ instead of $8x$ (forgot to add $3x$). Option C has $+12$ instead of $-12$. Option D uses $2x$ (perhaps $5x - 3x$) instead of correctly adding $5x + 3x = 8x$.

Answer: The peer is correct because the first student omitted the cross product $2(2x)(3) = 12x$.

Correct expansion: $(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9$. Why the correct answer works: Option C identifies the error: the first student omitted the middle term $12x$. The peer's expansion is correct. Why distractors fail: Option A is wrong because $4x^2 + 9 \neq 4x^2 + 12x + 9$. Option B is wrong because exponents do not distribute over addition: $(a + b)^2 \neq a^2 + b^2$. Option D gives $6x$ instead of $12x$, missing the factor of 2.

Answer: The student applied the exponent to each term individually but ignored the cross product $2 \cdot x \cdot 3 = 6x$.

Identify the common error: The student treated $(x + 3)^2$ as $x^2 + 3^2$, distributing the exponent to each term individually. This ignores the cross-product terms. Correct expansion: $(x + 3)^2 = x^2 + 2(x)(3) + 9 = x^2 + 6x + 9$. The missing $6x$ term is the key. Why distractors fail: Option A introduces subtraction with no basis. Option C confuses the constant ($3^2 = 9$, not $3 + 3 = 6$). Option D is factually wrong since $x^2 + 9 \neq (x+3)^2$.

Answer: It is distributed to every term inside the parentheses by multiplication.

Apply the distributive property: $3(x^2 - 2x + 5) = 3 \cdot x^2 + 3 \cdot (-2x) + 3 \cdot 5 = 3x^2 - 6x + 15$. Why the correct answer works: Option C correctly states that the factor outside the parentheses multiplies every term inside. Why distractors fail: Option A confuses distribution (multiplication) with addition. Option B incorrectly limits distribution to only some terms. Option D confuses a coefficient with an exponent.

Answer: $8x$

Expand each square: $(x + 2)^2 = x^2 + 4x + 4$ and $(x - 2)^2 = x^2 - 4x + 4$. Subtract: $(x^2 + 4x + 4) - (x^2 - 4x + 4) = x^2 + 4x + 4 - x^2 + 4x - 4 = 8x$. Why distractors fail: Option A ($0$) would only be true if both expansions were identical. Option C ($2x^2 + 8$) results from adding instead of subtracting. Option D ($8$) confuses the result with only the constant part.

Answer: First, Outer, Inner, Last

Recall the FOIL acronym: FOIL is a mnemonic for the four products needed when multiplying two binomials: First terms, Outer terms, Inner terms, and Last terms. Why the correct answer works: Option A correctly identifies all four words: First, Outer, Inner, Last. Why distractors fail: Options B, C, and D substitute incorrect words. 'Factor, Order, Identify, List' describes a different process. 'Opposite' and 'Least' are not part of the FOIL mnemonic.

Answer: The first student, because $-15x + 2x = -13x$.

Expand using FOIL: F: $3x \cdot 2x = 6x^2$. O: $3x \cdot (-5) = -15x$. I: $1 \cdot 2x = 2x$. L: $1 \cdot (-5) = -5$. Combine like terms: $-15x + 2x = -13x$, giving $6x^2 - 13x - 5$. Why distractors fail: Option B is wrong because the second student forgot to add the Inner product $+2x$. Option C is wrong because $1 \cdot (-5) = -5$, not $+5$. Option D is wrong because the two expressions are not equal ($-13x \neq -15x$).

Answer: Distributive property

Identify the property: The distributive property allows you to multiply a single factor across each term inside parentheses: $a(b + c) = ab + ac$. Why the correct answer works: Option C, the distributive property, is the exact name for this rule. Why distractors fail: The commutative property deals with order ($a + b = b + a$). The associative property deals with grouping ($(a + b) + c = a + (b + c)$). The identity property states that $a \cdot 1 = a$ or $a + 0 = a$.

Answer: $x \cdot 5 = 5x$

Identify the Outer terms: In $(x + 3)(x + 5)$, the 'Outer' terms are the first term of the first binomial ($x$) and the last term of the second binomial ($5$). Why the correct answer works: The outer product is $x \cdot 5 = 5x$. Why distractors fail: $x \cdot x = x^2$ is the First product. $3 \cdot x = 3x$ is the Inner product. $3 \cdot 5 = 15$ is the Last product.

Answer: $a^2 + 2ab + b^2$

Recall the squared binomial formula: $(a + b)^2$ means $(a + b)(a + b)$. Using FOIL: $a^2 + ab + ab + b^2 = a^2 + 2ab + b^2$. Why the correct answer works: Option B gives the complete expansion with the essential middle term $2ab$. Why distractors fail: Option A ($a^2 + b^2$) is the most common error — it omits the $2ab$ term. Option C has the wrong sign on $2ab$; that formula applies to $(a - b)^2$. Option D incorrectly treats the squaring as multiplication by 2.

Answer: $x^2 + 9x + 14$

Apply FOIL: First: $x \cdot x = x^2$. Outer: $x \cdot 7 = 7x$. Inner: $2 \cdot x = 2x$. Last: $2 \cdot 7 = 14$. Combine like terms: $x^2 + 7x + 2x + 14 = x^2 + 9x + 14$. Why distractors fail: Option A omits the $9x$ middle term. Option C uses $5x$ (which is $7 - 2$ instead of $7 + 2$). Option D has $9$ instead of $14$ for the constant ($2 + 7$ instead of $2 \times 7$).

Answer: $x^2 - 10x + 25$

Apply the squared binomial formula: $(x - 5)^2 = x^2 - 2(x)(5) + 5^2 = x^2 - 10x + 25$. Why the correct answer works: Option C correctly includes the middle term $-10x$ and the constant $+25$. Why distractors fail: Option A ($x^2 - 25$) confuses squaring a binomial with a difference of squares. Option B ($x^2 + 25$) omits the middle term entirely. Option D ($x^2 - 5x + 25$) uses $-5x$ instead of $-2(5)x = -10x$.

Answer: $2x^2 - 5x - 12$

Apply FOIL: First: $2x \cdot x = 2x^2$. Outer: $2x \cdot (-4) = -8x$. Inner: $3 \cdot x = 3x$. Last: $3 \cdot (-4) = -12$. Combine like terms: $2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$. Why distractors fail: Option B has $+5x$ instead of $-5x$ (sign error on the outer term). Option C has $-8x$ and failed to combine the inner term $+3x$. Option D has $+12$ instead of $-12$ (sign error on the last product).