what is the product of $a+3$ and $-2a^2+15a+6b^2$?

$-2a^3+9a^2+45a+24b^2$
$-2a^3+21a^2+45a+24b^2$
$-2a^3+9a^2+45a+6ab^2+18b^2$
$-2a^3+21a^2+45a+6ab^2+18b^2$

Question

what is the product of $a+3$ and $-2a^2+15a+6b^2$?

$-2a^3+9a^2+45a+24b^2$
$-2a^3+21a^2+45a+24b^2$
$-2a^3+9a^2+45a+6ab^2+18b^2$
$-2a^3+21a^2+45a+6ab^2+18b^2$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Multiplication", "Algebraic Simplification" ], "new_concepts": [], "current_concepts": [ "Polynomial Multiplication", "Algebraic Simplification" ] } ## Set up the product expression $$ (a + 3)(-2a^2 + 15a + 6b^2) $$ ## Distribute each term $$ \begin{aligned} &= a(-2a^2 + 15a + 6b^2) + 3(-2a^2 + 15a + 6b^2) \ &= -2a^3 + 15a^2 + 6ab^2 - 6a^2 + 45a + 18b^2 \end{aligned} $$ ## Combine like terms $$ \begin{aligned} &= -2a^3 + (15a^2 - 6a^2) + 45a + 6ab^2 + 18b^2 \ &= -2a^3 + 9a^2 + 45a + 6ab^2 + 18b^2 \end{aligned} $$

(A) $-2a^3+9a^2+45a+24b^2$

(B) $-2a^3+21a^2+45a+24b^2$

(C) $-2a^3+9a^2+45a+6ab^2+18b^2$

(D) $-2a^3+21a^2+45a+6ab^2+18b^2$

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

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QUESTION IMAGE

Question

Explanation:

Set up the product expression

Distribute each term

Combine like terms

Answer: