Question

Perform the indicated operation and simplify.

$\frac{4x}{x+2}·\frac{x^2+5x+6}{12x^2}$

Short answer

The solution is$\frac{x+3}{3x}$.

Step-by-step solution

Step 1. Given information.

The given rational expression is:
$\frac{4x}{x+2}·\frac{x^2+5x+6}{12x^2}$

Step 2. Simplify the rational expression.

$$\begin{gathered} \frac{4x}{x+2}·\frac{x^2+5x+6}{12x^2}=\frac{4x}{x+2}·\frac{x^2+3x+2x+6}{12x^2} \\ \frac{4x}{x+2}·\frac{x^2+5x+6}{12x^2}=\frac{4x}{x+2}·\frac{x\left(x+3\right)+2\left(x+3\right)}{12x^2} \\ \frac{4x}{x+2}·\frac{x^2+5x+6}{12x^2}=\frac{4x}{x+2}·\frac{(x+3)\left(x+2\right)}{4·3·x^2} \\ \frac{4x}{x+2}·\frac{x^2+5x+6}{12x^2}=\frac{x+3}{3x} \end{gathered}$$

Step 3. Simplified answer.

Hence, the simplified rational expression is$\frac{x+3}{3x}$.