Question

first up, lets review the assignments learning objectives
get familiar with this topic by reviewing instruction and answering a couple of questions.

question
simplify the rational expression: \\(\frac{3c + 6}{3c + 9} \div \frac{c^2 - 6c + 8}{c^2 + 2c - 3} \cdot \frac{3c - 6}{3c + 12}\\).

provide your answer below:

Explanation:

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<reasoning>

Factor all numerators and denominators

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

$$\begin{aligned} 3c + 6 &= 3(c + 2) \\ 3c + 9 &= 3(c + 3) \\ c^2 - 6c + 8 &= (c - 2)(c - 4) \\ c^2 + 2c - 3 &= (c + 3)(c - 1) \\ 3c - 6 &= 3(c - 2) \\ 3c + 12 &= 3(c + 4) \end{aligned}$$

\]

Rewrite division as multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{3(c + 2)}{3(c + 3)} \cdot \frac{(c + 3)(c - 1)}{(c - 2)(c - 4)} \cdot \frac{3(c - 2)}{3(c + 4)}
\]

Multiply the rational expressions

Using the Multiplying Rational Expressions knowledge point
\[
\frac{3(c + 2) \cdot (c + 3)(c - 1) \cdot 3(c - 2)}{3(c + 3) \cdot (c - 2)(c - 4) \cdot 3(c + 4)}
\]

Simplify by canceling common factors

Using the Simplifying Rational Expressions knowledge point
\[

$$\begin{aligned} &\frac{\cancel{3}(c + 2) \cdot \cancel{(c + 3)}(c - 1) \cdot \cancel{3}\cancel{(c - 2)}}{\cancel{3}\cancel{(c + 3)} \cdot \cancel{(c - 2)}(c - 4) \cdot \cancel{3}(c + 4)} \\ &= \frac{(c + 2)(c - 1)}{(c - 4)(c + 4)} \\ &= \frac{c^2 + c - 2}{c^2 - 16} \end{aligned}$$

\]
</reasoning>

<answer>
\[\frac{(c + 2)(c - 1)}{(c - 4)(c + 4)}\]
</answer>

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Answer:

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<reasoning>

Factor all numerators and denominators

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

$$\begin{aligned} 3c + 6 &= 3(c + 2) \\ 3c + 9 &= 3(c + 3) \\ c^2 - 6c + 8 &= (c - 2)(c - 4) \\ c^2 + 2c - 3 &= (c + 3)(c - 1) \\ 3c - 6 &= 3(c - 2) \\ 3c + 12 &= 3(c + 4) \end{aligned}$$

\]

Rewrite division as multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{3(c + 2)}{3(c + 3)} \cdot \frac{(c + 3)(c - 1)}{(c - 2)(c - 4)} \cdot \frac{3(c - 2)}{3(c + 4)}
\]

Multiply the rational expressions

Using the Multiplying Rational Expressions knowledge point
\[
\frac{3(c + 2) \cdot (c + 3)(c - 1) \cdot 3(c - 2)}{3(c + 3) \cdot (c - 2)(c - 4) \cdot 3(c + 4)}
\]

Simplify by canceling common factors

Using the Simplifying Rational Expressions knowledge point
\[

$$\begin{aligned} &\frac{\cancel{3}(c + 2) \cdot \cancel{(c + 3)}(c - 1) \cdot \cancel{3}\cancel{(c - 2)}}{\cancel{3}\cancel{(c + 3)} \cdot \cancel{(c - 2)}(c - 4) \cdot \cancel{3}(c + 4)} \\ &= \frac{(c + 2)(c - 1)}{(c - 4)(c + 4)} \\ &= \frac{c^2 + c - 2}{c^2 - 16} \end{aligned}$$

\]
</reasoning>

<answer>
\[\frac{(c + 2)(c - 1)}{(c - 4)(c + 4)}\]
</answer>

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