Question

In the following exercise, divide.

$\frac{8a^2+16a}{a-4}.\frac{a^2+2a-24}{a^2+7a+10}÷\frac{2a^2-6a}{a+5}$

Short answer

The simplified expression is

$\frac{4(a+6)}{(a-3)}$

Step-by-step solution

Step 1. Given information

expression to divide is

$\frac{8a^2+16a}{a-4}.\frac{a^2+2a-24}{a^2+7a+10}÷\frac{2a^2-6a}{a+5}$

Step 2. Rewrite as the product of first times the reciprocal of the second.   

$\frac{8a^2+16a}{a-4}.\frac{a^2+2a-24}{a^2+7a+10}.\frac{a+5}{2a^2-6a}$

Step 3. Factorize the denominator and numerator 

$$\begin{gathered} \frac{8a(a+2)}{a-4}.\frac{a^2+6a-4a-24}{a^2+5a+2a+10}.\frac{a+5}{2a(a-3)} \\ =\frac{8a(a+2)}{a-4}.\frac{a(a+6)-4(a+6)}{a(a+5)+2(a+5)}.\frac{a+5}{2a(a-3)} \\ =\frac{8a(a+2)}{a-4}.\frac{(a-4)(a+6)}{(a+2)(a+5)}.\frac{a+5}{2a(a-3)} \end{gathered}$$

Step 4.Multiply and remove common factors 

$$\begin{gathered} \frac{8a(a+2)(a-4)(a+6)(a+5)}{2a(a-4)(a+2)(a+5)(a-3)} \\ =\frac{4(a+6)}{a-3} \end{gathered}$$