Answer
$ \frac{10}{(x+2)(x-2)}$.
Work Step by Step
The given expression is
$=\frac{3}{x-2}-\frac{2}{x+2}-\frac{x}{x^2-4}$
Factor $x^2-4$.
$=x^2-2^2$
Use the algebraic expression $a^2-b^2=(a+b)(a-b)$.
$=(x+2)(x-2)$
Substitute the factor into the given expression.
$=\frac{3}{x-2}-\frac{2}{x+2}-\frac{x}{(x+2)(x-2)}$
LCM of all the denominators $=(x+2)(x-2)$
$=\frac{(x+2)}{(x+2)} \cdot \frac{3}{x-2}-\frac{(x-2)}{(x-2)} \cdot \frac{2}{x+2}-\frac{x}{(x+2)(x-2)}$
Simplify.
$= \frac{3(x+2)}{(x+2)(x-2)}-\frac{2(x-2)}{(x+2)(x-2)}-\frac{x}{(x+2)(x-2)}$
$= \frac{3(x+2)-2(x-2)-x}{(x+2)(x-2)}$
Simplify.
$= \frac{3x+6-2x+4-x}{(x+2)(x-2)}$
$= \frac{10}{(x+2)(x-2)}$.
