Question

what is the factored form of the polynomial?

(x^2 + 9x + 20)

• ((x - 4)(x - 5))
• ((x - 2)(x - 10))
• ((x + 4)(x + 5))
• ((x + 2)(x + 10))

Explanation:

<pre_analysis>
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<reasoning>

Identify the coefficients of the quadratic trinomial

\[

$$\begin{aligned} &x^2 + 9x + 20\\ &a = 1,\quad b = 9,\quad c = 20 \end{aligned}$$

\]

Find two numbers that multiply to c and add to b

\[

$$\begin{aligned} &p \cdot q = 20\\ &p + q = 9\\ &\text{Factors of } 20: (1, 20), (2, 10), (4, 5)\\ &4 + 5 = 9 \implies p = 4,\quad q = 5 \end{aligned}$$

\]

Write the trinomial in its factored form

\[
x^2 + 9x + 20 = (x + 4)(x + 5)
\]
</reasoning>

<answer>
<mcq-option>(A) \((x - 4)(x - 5)\)</mcq-option>
<mcq-option>(B) \((x - 2)(x - 10)\)</mcq-option>
<mcq-correct>(C) \((x + 4)(x + 5)\)</mcq-correct>
<mcq-option>(D) \((x + 2)(x + 10)\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Mathematics",
"Algebra",
"Factoring Trinomials"
]
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</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
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</pre_analysis>

<reasoning>

Identify the coefficients of the quadratic trinomial

\[

$$\begin{aligned} &x^2 + 9x + 20\\ &a = 1,\quad b = 9,\quad c = 20 \end{aligned}$$

\]

Find two numbers that multiply to c and add to b

\[

$$\begin{aligned} &p \cdot q = 20\\ &p + q = 9\\ &\text{Factors of } 20: (1, 20), (2, 10), (4, 5)\\ &4 + 5 = 9 \implies p = 4,\quad q = 5 \end{aligned}$$

\]

Write the trinomial in its factored form

\[
x^2 + 9x + 20 = (x + 4)(x + 5)
\]
</reasoning>

<answer>
<mcq-option>(A) \((x - 4)(x - 5)\)</mcq-option>
<mcq-option>(B) \((x - 2)(x - 10)\)</mcq-option>
<mcq-correct>(C) \((x + 4)(x + 5)\)</mcq-correct>
<mcq-option>(D) \((x + 2)(x + 10)\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
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"Mathematics",
"Algebra",
"Factoring Trinomials"
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