Question

In the following exercises, divide the rational expressions.

$\frac{{\displaystyle \frac{2a^2-a-21}{5a+20}}}{{\displaystyle \frac{a^2+7a+12}{a^2+8a+16}}}$

Short answer

Simplified expression is$=\frac{\left(2a-7\right)}{5}.$

Step-by-step solution

Step 1. Given Information

$=\frac{2a^2-a-21}{5a+20}÷\frac{a^2+7a+12}{a^2+8a+16}$

Step 2. Rewrite as product of first times reciprocalof second.

$=\frac{2a^2-a-21}{5a+20}.\frac{a^2+8a+16}{a^2+7a+12}$

Step 3. Factor the numerators and thedenominators, and then multiply.

$$\begin{gathered} =\frac{\left(2a-7\right)\left(a+3\right)}{5(a+4)}.\frac{\left(a+4\right)\left(a+4\right)}{\left(a+4\right)\left(a+3\right)} \\ =\frac{\left(2a-7\right)\left(a+3\right)\left(a+4\right)\left(a+4\right)}{5(a+4)\left(a+4\right)\left(a+3\right)} \end{gathered}$$

Step 4. Simplify by dividing out common factors.

$=\frac{\left(2a-7\right)}{5}$