Question

which expression is the factorization of \\(x^2 + 10x + 21\\)?

\\((x + 3)(x + 7)\\)
\\((x + 4)(x + 6)\\)
\\((x + 6)(x + 15)\\)
\\((x + 7)(x + 14)\\)

Explanation:

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<reasoning>

Identify coefficients of the quadratic trinomial

\[
x^2 + 10x + 21 \implies a = 1, \quad b = 10, \quad c = 21
\]

Find two numbers that multiply to 21 and add to 10

\[

$$\begin{aligned} &p \cdot q = 21 \\ &p + q = 10 \\ &\implies p = 3, \quad q = 7 \end{aligned}$$

\]

Write the factored form

\[
x^2 + 10x + 21 = (x + 3)(x + 7)
\]
</reasoning>

<answer>
<mcq-correct>(A) \((x + 3)(x + 7)\)</mcq-correct>
<mcq-option>(B) \((x + 4)(x + 6)\)</mcq-option>
<mcq-option>(C) \((x + 6)(x + 15)\)</mcq-option>
<mcq-option>(D) \((x + 7)(x + 14)\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify coefficients of the quadratic trinomial

\[
x^2 + 10x + 21 \implies a = 1, \quad b = 10, \quad c = 21
\]

Find two numbers that multiply to 21 and add to 10

\[

$$\begin{aligned} &p \cdot q = 21 \\ &p + q = 10 \\ &\implies p = 3, \quad q = 7 \end{aligned}$$

\]

Write the factored form

\[
x^2 + 10x + 21 = (x + 3)(x + 7)
\]
</reasoning>

<answer>
<mcq-correct>(A) \((x + 3)(x + 7)\)</mcq-correct>
<mcq-option>(B) \((x + 4)(x + 6)\)</mcq-option>
<mcq-option>(C) \((x + 6)(x + 15)\)</mcq-option>
<mcq-option>(D) \((x + 7)(x + 14)\)</mcq-option>
</answer>

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