Question

current objective
multiply rational expressions

question
multiply: \\(\frac{q^2 + 5q + 6}{6q + 18} \cdot \frac{9q + 27}{q^2 + 8q + 15}\\)

note: enter answer in factored form

provide your answer below:

Explanation:

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<reasoning>

Factor the numerators

Using the Factoring Trinomials and Greatest Common Factor Factoring knowledge points
\[

$$\begin{aligned} q^2 + 5q + 6 &= (q + 2)(q + 3) \\ 9q + 27 &= 9(q + 3) \end{aligned}$$

\]

Factor the denominators

Using the Factoring Trinomials and Greatest Common Factor Factoring knowledge points
\[

$$\begin{aligned} 6q + 18 &= 6(q + 3) \\ q^2 + 8q + 15 &= (q + 3)(q + 5) \end{aligned}$$

\]

Multiply the rational expressions

Using the Multiplying Rational Expressions knowledge point
\[
\frac{q^2 + 5q + 6}{6q + 18} \cdot \frac{9q + 27}{q^2 + 8q + 15} = \frac{(q + 2)(q + 3)}{6(q + 3)} \cdot \frac{9(q + 3)}{(q + 3)(q + 5)}
\]

Simplify the combined expression

Using the Multiplying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{9(q + 2)(q + 3)(q + 3)}{6(q + 3)(q + 3)(q + 5)} &= \frac{9(q + 2)}{6(q + 5)} \\ &= \frac{3(q + 2)}{2(q + 5)} \end{aligned}$$

\]
</reasoning>

<answer>
Multiply: \(\frac{q^2 + 5q + 6}{6q + 18} \cdot \frac{9q + 27}{q^2 + 8q + 15}\)

Provide your answer below:
<blank>\(\frac{3(q+2)}{2(q+5)}\)</blank>
</answer>

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</post_analysis>

Answer:

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<reasoning>

Factor the numerators

Using the Factoring Trinomials and Greatest Common Factor Factoring knowledge points
\[

$$\begin{aligned} q^2 + 5q + 6 &= (q + 2)(q + 3) \\ 9q + 27 &= 9(q + 3) \end{aligned}$$

\]

Factor the denominators

Using the Factoring Trinomials and Greatest Common Factor Factoring knowledge points
\[

$$\begin{aligned} 6q + 18 &= 6(q + 3) \\ q^2 + 8q + 15 &= (q + 3)(q + 5) \end{aligned}$$

\]

Multiply the rational expressions

Using the Multiplying Rational Expressions knowledge point
\[
\frac{q^2 + 5q + 6}{6q + 18} \cdot \frac{9q + 27}{q^2 + 8q + 15} = \frac{(q + 2)(q + 3)}{6(q + 3)} \cdot \frac{9(q + 3)}{(q + 3)(q + 5)}
\]

Simplify the combined expression

Using the Multiplying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{9(q + 2)(q + 3)(q + 3)}{6(q + 3)(q + 3)(q + 5)} &= \frac{9(q + 2)}{6(q + 5)} \\ &= \frac{3(q + 2)}{2(q + 5)} \end{aligned}$$

\]
</reasoning>

<answer>
Multiply: \(\frac{q^2 + 5q + 6}{6q + 18} \cdot \frac{9q + 27}{q^2 + 8q + 15}\)

Provide your answer below:
<blank>\(\frac{3(q+2)}{2(q+5)}\)</blank>
</answer>

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