select the correct answer.

consider these shapes.

shape a
width: $3x^2 + 2$
length: $7x^2 + 4x + 8$

shape b
side: $3x^2 + 2$

if shape b is a square, which polynomial represents the difference of the areas of shape a and shape b?

$8x^2 + 8x + 12$
$32x^2 + 8x + 28$
$30x^4 + 12x^3 + 40x^2 + 8x + 20$
$12x^4 + 12x^3 + 26x^2 + 8x + 12$

Question

select the correct answer.

consider these shapes.

shape a
width: $3x^2 + 2$
length: $7x^2 + 4x + 8$

shape b
side: $3x^2 + 2$

if shape b is a square, which polynomial represents the difference of the areas of shape a and shape b?

$8x^2 + 8x + 12$
$32x^2 + 8x + 28$
$30x^4 + 12x^3 + 40x^2 + 8x + 20$
$12x^4 + 12x^3 + 26x^2 + 8x + 12$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Polynomial Operations", "Geometric Modeling" ], "current_concepts": [ "Polynomial Operations", "Geometric Modeling", "Algebraic Modeling" ] } ## Calculate the area of Shape A and Shape B $$ \begin{aligned} \text{Area}_A &= (3x^2 + 2)(7x^2 + 4x + 8) \ &= 21x^4 + 12x^3 + 24x^2 + 14x^2 + 8x + 16 \ &= 21x^4 + 12x^3 + 38x^2 + 8x + 16 \end{aligned} $$ $$ \begin{aligned} \text{Area}_B &= (3x^2 + 2)^2 \ &= 9x^4 + 12x^2 + 4 \end{aligned} $$ ## Subtract the area of Shape B from Shape A $$ \begin{aligned} \text{Difference} &= \text{Area}_A - \text{Area}_B \ &= (21x^4 + 12x^3 + 38x^2 + 8x + 16) - (9x^4 + 12x^2 + 4) \ &= (21 - 9)x^4 + 12x^3 + (38 - 12)x^2 + 8x + (16 - 4) \ &= 12x^4 + 12x^3 + 26x^2 + 8x + 12 \end{aligned} $$

(A) $8x^2 + 8x + 12$

(B) $32x^2 + 8x + 28$

(C) $30x^4 + 12x^3 + 40x^2 + 8x + 20$

(D) $12x^4 + 12x^3 + 26x^2 + 8x + 12$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Operations" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the area of Shape A and Shape B

Subtract the area of Shape B from Shape A

Answer: