Question

In the following exercise, perform the indicated operation:

$\frac{3d}{d+2}+\frac{4}{d}-\frac{d+8}{d^2+2d}$

Short answer

$\frac{3\left(d+1\right)}{\left(d+2\right)}$

Step-by-step solution

Step 1. Given information.

The following expression is given:

$\frac{3d}{d+2}+\frac{4}{d}-\frac{d+8}{d^2+2d}$

Step 2. Perform the indicated operation.

Factorise the denominators:

$\frac{3d}{d+2}+\frac{4}{d}-\frac{d+8}{d\left(d+2\right)}$

Find out the LCD:

$d(d+2)$

Rewrite the expression with denominators equal to the LCD:

$\frac{3d.d}{d\left(d+2\right)}+\frac{4\left(d+2\right)}{d\left(d+2\right)}-\frac{d+8}{d\left(d+2\right)}$

Simplify:

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

Cancel the common factors:

$\frac{3\left(d+1\right)}{\left(d+2\right)}$