Question

Divide:$\frac{{\displaystyle \frac{3x^2+7x+2}{4x+24}}}{{\displaystyle \frac{3x^2-14x-5}{x^2+x-30}}}$.

Short answer

The required answer is$\frac{x+2}{4}$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$\frac{{\displaystyle \frac{3x^2+7x+2}{4x+24}}}{{\displaystyle \frac{3x^2-14x-5}{x^2+x-30}}}$

Step 2. Simplify the expression.

$\frac{{\displaystyle \frac{3x^2+7x+2}{4x+24}}}{{\displaystyle \frac{3x^2-14x-5}{x^2+x-30}}}$

Rewrite with division sign,

$\frac{3x^2+7x+2}{4x+24}÷\frac{3x^2-14x-5}{x^2+x-30}$

Rewrite the division as the product of the first rational expression and the reciprocal of the second.

$\frac{3x^2+7x+2}{4x+24}·\frac{x^2+x-30}{3x^2-14x-5}$

Factor the numerators and denominators and then multiply.

$$\begin{gathered} \frac{\left(3x+1\right)\left(x+2\right)}{4\left(x+6\right)}·\frac{\left(x-5\right)\left(x+6\right)}{\left(3x+1\right)\left(x-5\right)} \\ \frac{\left(3x+1\right)\left(x+2\right)\left(x-5\right)\left(x+6\right)}{4\left(x+6\right)\left(3x+1\right)\left(x-5\right)} \end{gathered}$$

Simplify by dividing out common factors.

$\frac{x+2}{4}$