Question

Divide Rational Expressions:$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{2d^2\hspace{0.17em}+\hspace{0.17em}d\hspace{0.17em}-\hspace{0.17em}3}{d^2\hspace{0.17em}-\hspace{0.17em}16}÷\frac{2d^2\hspace{0.17em}-\hspace{0.17em}9d\hspace{0.17em}-\hspace{0.17em}18}{d^2\hspace{0.17em}-\hspace{0.17em}8d\hspace{0.17em}+\hspace{0.17em}16}$.

Short answer

The required answer is$\frac{\left(d-1\right)\left(d-4\right)}{\left(d+4\right)\left(d-6\right)}$.

Step-by-step solution

Step 1. Given information.

The given expression is$\frac{2d^2\hspace{0.17em}+\hspace{0.17em}d\hspace{0.17em}-\hspace{0.17em}3}{d2\hspace{0.17em}-\hspace{0.17em}16}÷\frac{2d^2\hspace{0.17em}-\hspace{0.17em}9d\hspace{0.17em}-\hspace{0.17em}18}{d^2\hspace{0.17em}-\hspace{0.17em}8d\hspace{0.17em}+\hspace{0.17em}16}$.

Step 2.  Simplify the expression. 

$$\begin{gathered} \frac{2d^2\hspace{0.17em}+\hspace{0.17em}d\hspace{0.17em}-\hspace{0.17em}3}{d2\hspace{0.17em}-\hspace{0.17em}16}÷\frac{2d^2\hspace{0.17em}-\hspace{0.17em}9d\hspace{0.17em}-\hspace{0.17em}18}{d^2\hspace{0.17em}-\hspace{0.17em}8d\hspace{0.17em}+\hspace{0.17em}16} \\ =\frac{\frac{2d^2+d-3}{d^2-16}}{\frac{2d^2-9d-18}{d^2-8d+16}} \end{gathered}$$

Apply the fraction rule.

$$\begin{gathered} \frac{\left(2d^2+d-3\right)\left(d^2-8d+16\right)}{\left(d^2-16\right)\left(2d^2-9d-18\right)} \\ =\frac{\left(d-1\right)\left(2d+3\right)\left(d^2-8d+16\right)}{\left(d^2-16\right)\left(2d+3\right)\left(d-6\right)} \\ =\frac{\left(d-1\right){\left(d-4\right)}^2}{\left(d+4\right)\left(d-4\right)\left(d-6\right)} \\ =\frac{\left(d-1\right)\left(d-4\right)\left(d-4\right)}{\left(d+4\right)\left(d-4\right)\left(d-6\right)} \\ =\frac{\left(d-1\right)\left(d-4\right)}{\left(d+4\right)\left(d-6\right)} \end{gathered}$$