Question

question

simplify the rational expression: \\(\frac{4m + 4}{3m - 15} \cdot \frac{m^2 - 3m - 10}{m^2 - 4m - 32} \div \frac{12m - 36}{6m - 48}

provide your answer below:

Explanation:

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<reasoning>

Convert division to multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{4m + 4}{3m - 15} \cdot \frac{m^2 - 3m - 10}{m^2 - 4m - 32} \div \frac{12m - 36}{6m - 48} = \frac{4m + 4}{3m - 15} \cdot \frac{m^2 - 3m - 10}{m^2 - 4m - 32} \cdot \frac{6m - 48}{12m - 36}
\]

Factor all linear terms

Using the Greatest Common Factor Factoring knowledge point
\[

$$\begin{aligned} 4m + 4 &= 4(m + 1) \\ 3m - 15 &= 3(m - 5) \\ 6m - 48 &= 6(m - 8) \\ 12m - 36 &= 12(m - 3) \end{aligned}$$

\]

Factor all quadratic trinomials

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} m^2 - 3m - 10 &= (m - 5)(m + 2) \\ m^2 - 4m - 32 &= (m - 8)(m + 4) \end{aligned}$$

\]

Multiply and simplify the expression

Using the Multiplying Rational Expressions and Simplifying Rational Expressions knowledge points
\[

$$\begin{aligned} &\frac{4(m + 1)}{3(m - 5)} \cdot \frac{(m - 5)(m + 2)}{(m - 8)(m + 4)} \cdot \frac{6(m - 8)}{12(m - 3)} \\ &= \frac{4 \cdot 6 \cdot (m + 1)(m - 5)(m + 2)(m - 8)}{3 \cdot 12 \cdot (m - 5)(m - 8)(m + 4)(m - 3)} \\ &= \frac{24}{36} \cdot \frac{(m + 1)(m + 2)}{(m + 4)(m - 3)} \\ &= \frac{2(m + 1)(m + 2)}{3(m + 4)(m - 3)} \end{aligned}$$

\]
</reasoning>

<answer>
\[\frac{2(m + 1)(m + 2)}{3(m + 4)(m - 3)}\]
</answer>

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Answer:

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<reasoning>

Convert division to multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{4m + 4}{3m - 15} \cdot \frac{m^2 - 3m - 10}{m^2 - 4m - 32} \div \frac{12m - 36}{6m - 48} = \frac{4m + 4}{3m - 15} \cdot \frac{m^2 - 3m - 10}{m^2 - 4m - 32} \cdot \frac{6m - 48}{12m - 36}
\]

Factor all linear terms

Using the Greatest Common Factor Factoring knowledge point
\[

$$\begin{aligned} 4m + 4 &= 4(m + 1) \\ 3m - 15 &= 3(m - 5) \\ 6m - 48 &= 6(m - 8) \\ 12m - 36 &= 12(m - 3) \end{aligned}$$

\]

Factor all quadratic trinomials

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} m^2 - 3m - 10 &= (m - 5)(m + 2) \\ m^2 - 4m - 32 &= (m - 8)(m + 4) \end{aligned}$$

\]

Multiply and simplify the expression

Using the Multiplying Rational Expressions and Simplifying Rational Expressions knowledge points
\[

$$\begin{aligned} &\frac{4(m + 1)}{3(m - 5)} \cdot \frac{(m - 5)(m + 2)}{(m - 8)(m + 4)} \cdot \frac{6(m - 8)}{12(m - 3)} \\ &= \frac{4 \cdot 6 \cdot (m + 1)(m - 5)(m + 2)(m - 8)}{3 \cdot 12 \cdot (m - 5)(m - 8)(m + 4)(m - 3)} \\ &= \frac{24}{36} \cdot \frac{(m + 1)(m + 2)}{(m + 4)(m - 3)} \\ &= \frac{2(m + 1)(m + 2)}{3(m + 4)(m - 3)} \end{aligned}$$

\]
</reasoning>

<answer>
\[\frac{2(m + 1)(m + 2)}{3(m + 4)(m - 3)}\]
</answer>

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