Question

For the following exercises, perform the indicated operations.$\frac{10m^2+80m}{3m-9}.\frac{m^2+4m-21}{m^2-9m+20}÷\frac{5m^2+10m}{2m-10}$

Short answer

The simplified expression is$=\frac{4\left(m+8\right)\left(m+7\right)}{3\left(m-4\right)\left(m+2\right)}$

Step-by-step solution

Step 1. Given Information

$\frac{10m^2+80m}{3m-9}.\frac{m^2+4m-21}{m^2-9m+20}÷\frac{5m^2+10m}{2m-10}$

$=\frac{10m^2+80m}{3m-9}.\frac{m^2+4m-21}{m^2-9m+20}.\frac{2m-10}{5m^2+10m}$

Step 2. Factor the numerator and denominator

$$\begin{gathered} \frac{10m^2+80m}{3m-9}.\frac{m^2+4m-21}{m^2-9m+20}.\frac{2m-10}{5m^2+10m} \\ =\frac{10m\left(m+8\right)}{3\left(m-3\right)}.\frac{\left(m+7\right)\left(m-3\right)}{\left(m-4\right)\left(m-5\right)}.\frac{2\left(m-5\right)}{5m\left(m+2\right)} \end{gathered}$$

Step 3. Multiply numerators and denominators

$=\frac{2\times 10m\left(m+8\right)\left(m+7\right)\left(m-3\right)\left(m-5\right)}{3\times 5m\left(m-3\right)\left(m-4\right)\left(m-5\right)\left(m+2\right)}.$

Step  4. Remove common factors and simplify

$=\frac{4\left(m+8\right)\left(m+7\right)}{3\left(m-4\right)\left(m+2\right)}$