Question

In the following exercises, multiply the rational expressions.$\frac{72m-12m^2}{8m+32}.\frac{m^2+10m+24}{m^2-36}$

Short answer

The simplified answer is$\frac{-3m}{2}.$

Step-by-step solution

Step 1. Given Information

$\frac{72m-12m^2}{8m+32}.\frac{m^2+10m+24}{m^2-36}$

Step 2. Factor each numerator and denominator completely.

$72m-12m^2=12m\left(6-m\right)$and

$m^2+10m+24=\left(m+6\right)\left(m+4\right)$and

$8m+32=8\left(m+4\right)$and

$m^2-36=\left(m-6\right)\left(m+6\right)$

Step 3. Multiply the numerators and denominators.Simplify by dividing out common factors

$$\begin{gathered} \frac{72m-12m^2}{8m+32}.\frac{m^2+10m+24}{m^2-36} \\ =\frac{12m\left(6-m\right)}{8\left(m+4\right)}.\frac{\left(m+6\right)\left(m+4\right)}{\left(m-6\right)\left(m+6\right)} \end{gathered}$$

$$\begin{gathered} \frac{-12m\left(m-6\right)}{8\left(m+4\right)}.\frac{\left(m+6\right)\left(m+4\right)}{\left(m-6\right)\left(m+6\right)} \\ =\frac{-3m}{2} \end{gathered}$$