Question

select the correct answer.
which expression shows the factors of $5x^2 + 17x + 6$?

a. $(5x + 1)(x + 6)$

b. $(5x + 2)(x + 3)$

c. $(5x + 3)(x + 2)$

d. $(5x + 6)(x + 1)$

Explanation:

Step1: Expand Option A

To expand \((5x + 1)(x + 6)\), we use the distributive property (FOIL method):
\[

$$\begin{align*} (5x + 1)(x + 6)&=5x\cdot x+5x\cdot 6 + 1\cdot x+1\cdot 6\\ &=5x^{2}+30x+x + 6\\ &=5x^{2}+31x + 6 \end{align*}$$

\]
This is not equal to \(5x^{2}+17x + 6\), so Option A is incorrect.

Step2: Expand Option B

Expand \((5x + 2)(x + 3)\) using the FOIL method:
\[

$$\begin{align*} (5x + 2)(x + 3)&=5x\cdot x+5x\cdot 3+2\cdot x + 2\cdot 3\\ &=5x^{2}+15x+2x + 6\\ &=5x^{2}+17x + 6 \end{align*}$$

\]
This matches the given quadratic expression \(5x^{2}+17x + 6\). But we can check the other options for completeness.

Step3: Expand Option C

Expand \((5x + 3)(x + 2)\):
\[

$$\begin{align*} (5x + 3)(x + 2)&=5x\cdot x+5x\cdot 2+3\cdot x+3\cdot 2\\ &=5x^{2}+10x + 3x+6\\ &=5x^{2}+13x + 6 \end{align*}$$

\]
This is not equal to \(5x^{2}+17x + 6\), so Option C is incorrect.

Step4: Expand Option D

Expand \((5x + 6)(x + 1)\):
\[

$$\begin{align*} (5x + 6)(x + 1)&=5x\cdot x+5x\cdot 1+6\cdot x+6\cdot 1\\ &=5x^{2}+5x+6x + 6\\ &=5x^{2}+11x + 6 \end{align*}$$

\]
This is not equal to \(5x^{2}+17x + 6\), so Option D is incorrect.

Answer:

B. \((5x + 2)(x + 3)\)