Question

which expression is a factor of (10x^2 + 11x + 3)?

• (x - 3)
• (2x + 1)
• (5x - 3)
• (10x + 3)

Explanation:

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<reasoning>

Identify the coefficients of the quadratic trinomial

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} &a = 10\\ &b = 11\\ &c = 3 \end{aligned}$$

\]

Find two numbers that multiply to ac and add to b

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} &a \cdot c = 10 \cdot 3 = 30\\ &p \cdot q = 30,\quad p + q = 11\\ &p = 5,\quad q = 6 \end{aligned}$$

\]

Rewrite the middle term and factor by grouping

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} 10x^2 + 11x + 3 &= 10x^2 + 5x + 6x + 3\\ &= 5x(2x + 1) + 3(2x + 1)\\ &= (2x + 1)(5x + 3) \end{aligned}$$

\]

Compare the factors with the given options

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} &\text{Factors: } (2x + 1), (5x + 3)\\ &\text{Matching option: } 2x + 1 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x - 3\)</mcq-option>
<mcq-correct>(B) \(2x + 1\)</mcq-correct>
<mcq-option>(C) \(5x - 3\)</mcq-option>
<mcq-option>(D) \(10x + 3\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the coefficients of the quadratic trinomial

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} &a = 10\\ &b = 11\\ &c = 3 \end{aligned}$$

\]

Find two numbers that multiply to ac and add to b

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} &a \cdot c = 10 \cdot 3 = 30\\ &p \cdot q = 30,\quad p + q = 11\\ &p = 5,\quad q = 6 \end{aligned}$$

\]

Rewrite the middle term and factor by grouping

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} 10x^2 + 11x + 3 &= 10x^2 + 5x + 6x + 3\\ &= 5x(2x + 1) + 3(2x + 1)\\ &= (2x + 1)(5x + 3) \end{aligned}$$

\]

Compare the factors with the given options

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} &\text{Factors: } (2x + 1), (5x + 3)\\ &\text{Matching option: } 2x + 1 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x - 3\)</mcq-option>
<mcq-correct>(B) \(2x + 1\)</mcq-correct>
<mcq-option>(C) \(5x - 3\)</mcq-option>
<mcq-option>(D) \(10x + 3\)</mcq-option>
</answer>

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