Question

Divide Rational Expressions:$\frac{\frac{2a^2\hspace{0.17em}-\hspace{0.17em}a\hspace{0.17em}-\hspace{0.17em}21}{5a\hspace{0.17em}+\hspace{0.17em}20}}{\frac{a^2\hspace{0.17em}+\hspace{0.17em}7a\hspace{0.17em}+\hspace{0.17em}12}{a^2\hspace{0.17em}+\hspace{0.17em}8a\hspace{0.17em}+\hspace{0.17em}16}}$

Short answer

The required answer is$\frac{2a-7}{5}$.

Step-by-step solution

Step 1. Given information. 

The given expression is$\frac{\frac{2a^2-a-21}{5a+20}}{\frac{a^2+7a+12}{a^2+8a+16}}$

Step 2.  Simplify the expression.  

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

Apply the fraction rule.

$$\begin{gathered} =\frac{\left(2a^2-a-21\right)\left(a^2+8a+16\right)}{\left(5a+20\right)\left(a^2+7a+12\right)} \\ =\frac{\left(a+3\right)\left(2a-7\right)\left(a^2+8a+16\right)}{\left(5a+20\right)\left(a+3\right)\left(a+4\right)} \\ =\frac{\left(2a-7\right)\left(a^2+8a+16\right)}{\left(5a+20\right)\left(a+4\right)} \\ =\frac{\left(2a-7\right){\left(a+4\right)}^2}{\left(5a+20\right)\left(a+4\right)} \\ =\frac{\left(2a-7\right)\left(a+4\right)\left(a+4\right)}{\left(5a+20\right)\left(a+4\right)} \\ =\frac{\left(2a-7\right)\left(a+4\right)}{5a+20} \\ =\frac{\left(2a-7\right)\left(a+4\right)}{5\left(a+4\right)} \\ =\frac{2a-7}{5} \end{gathered}$$