Question

current objective
multiply and divide more than two rational expressions

question

simplify the rational expression: \\(\frac{3d + 9}{2d + 2} \cdot \frac{d^2 + 5d + 4}{d^2 - 4} \div \frac{3d - 9}{2d + 4}\\)

provide your answer below:

Explanation:

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<reasoning>

Convert division to multiplication

Using the Multiplying and Dividing Multiple Rational Expressions knowledge point
\[
\frac{3d+9}{2d+2} \cdot \frac{d^2+5d+4}{d^2-4} \div \frac{3d-9}{2d+4} = \frac{3d+9}{2d+2} \cdot \frac{d^2+5d+4}{d^2-4} \cdot \frac{2d+4}{3d-9}
\]

Factor the linear terms

Using the Greatest Common Factor Factoring knowledge point
\[

$$\begin{aligned} 3d + 9 &= 3(d + 3) \\ 2d + 2 &= 2(d + 1) \\ 2d + 4 &= 2(d + 2) \\ 3d - 9 &= 3(d - 3) \end{aligned}$$

\]

Factor the quadratic terms

Using the Factoring Trinomials and Difference of Squares knowledge points
\[

$$\begin{aligned} d^2 + 5d + 4 &= (d + 1)(d + 4) \\ d^2 - 4 &= (d - 2)(d + 2) \end{aligned}$$

\]

Substitute and simplify the expression

Using the Multiplying and Dividing Multiple Rational Expressions knowledge point
\[

$$\begin{aligned} &\frac{3(d+3)}{2(d+1)} \cdot \frac{(d+1)(d+4)}{(d-2)(d+2)} \cdot \frac{2(d+2)}{3(d-3)} \\ &= \frac{3 \cdot (d+3) \cdot (d+1) \cdot (d+4) \cdot 2 \cdot (d+2)}{2 \cdot (d+1) \cdot (d-2) \cdot (d+2) \cdot 3 \cdot (d-3)} \\ &= \frac{(d+3)(d+4)}{(d-2)(d-3)} \end{aligned}$$

\]
</reasoning>

<answer>
Simplify the rational expression: <blank>\(\frac{(d+3)(d+4)}{(d-2)(d-3)}\)</blank>
</answer>

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</post_analysis>

Answer:

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<reasoning>

Convert division to multiplication

Using the Multiplying and Dividing Multiple Rational Expressions knowledge point
\[
\frac{3d+9}{2d+2} \cdot \frac{d^2+5d+4}{d^2-4} \div \frac{3d-9}{2d+4} = \frac{3d+9}{2d+2} \cdot \frac{d^2+5d+4}{d^2-4} \cdot \frac{2d+4}{3d-9}
\]

Factor the linear terms

Using the Greatest Common Factor Factoring knowledge point
\[

$$\begin{aligned} 3d + 9 &= 3(d + 3) \\ 2d + 2 &= 2(d + 1) \\ 2d + 4 &= 2(d + 2) \\ 3d - 9 &= 3(d - 3) \end{aligned}$$

\]

Factor the quadratic terms

Using the Factoring Trinomials and Difference of Squares knowledge points
\[

$$\begin{aligned} d^2 + 5d + 4 &= (d + 1)(d + 4) \\ d^2 - 4 &= (d - 2)(d + 2) \end{aligned}$$

\]

Substitute and simplify the expression

Using the Multiplying and Dividing Multiple Rational Expressions knowledge point
\[

$$\begin{aligned} &\frac{3(d+3)}{2(d+1)} \cdot \frac{(d+1)(d+4)}{(d-2)(d+2)} \cdot \frac{2(d+2)}{3(d-3)} \\ &= \frac{3 \cdot (d+3) \cdot (d+1) \cdot (d+4) \cdot 2 \cdot (d+2)}{2 \cdot (d+1) \cdot (d-2) \cdot (d+2) \cdot 3 \cdot (d-3)} \\ &= \frac{(d+3)(d+4)}{(d-2)(d-3)} \end{aligned}$$

\]
</reasoning>

<answer>
Simplify the rational expression: <blank>\(\frac{(d+3)(d+4)}{(d-2)(d-3)}\)</blank>
</answer>

<post_analysis>
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"Mathematics",
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]
}
</post_analysis>