Question

Divide Rational Expressions:$\frac{\frac{4d^2\hspace{0.17em}+\hspace{0.17em}7d\hspace{0.17em}-\hspace{0.17em}2}{35d\hspace{0.17em}+\hspace{0.17em}10}}{\frac{d^2\hspace{0.17em}-\hspace{0.17em}4}{7d^2\hspace{0.17em}-\hspace{0.17em}12d\hspace{0.17em}-\hspace{0.17em}4}}$

Short answer

The required answer is$\frac{4d-1}{5}$.

Step-by-step solution

Step 1. Given information.

The given expression is$\frac{\frac{4d^2+7d-2}{35d+10}}{\frac{d^2-4}{7d^2-12d-4}}$.

Step 2.  Simplify the expression.  

$\frac{\frac{4d^2+7d-2}{35d+10}}{\frac{d^2-4}{7d^2-12d-4}}$

Apply the fraction rule.

$$\begin{gathered} \frac{\left(4d^2+7d-2\right)\left(7d^2-12d-4\right)}{\left(35d+10\right)\left(d^2-4\right)} \\ =\frac{\left(4d-1\right)\left(d+2\right)\left(7d^2-12d-4\right)}{\left(35d+10\right)\left(d+2\right)\left(d-2\right)} \\ =\frac{\left(4d-1\right)\left(7d^2-12d-4\right)}{\left(35d+10\right)\left(d-2\right)} \\ =\frac{\left(4d-1\right)\left(7d+2\right)\left(d-2\right)}{\left(35d+10\right)\left(d-2\right)} \\ =\frac{\left(4d-1\right)\left(7d+2\right)}{35d+10} \\ =\frac{\left(4d-1\right)\left(7d+2\right)}{5\left(7d+2\right)} \\ =\frac{4d-1}{5} \end{gathered}$$