Answer
$-\frac{11x+2}{(x-2)(x+2)}$
Work Step by Step
Since the least common multiple (LCM) of the denominator is $(x-2)(x+2)$, then we have
$$
\frac{x-3}{x+2}-\frac{x+4}{x-2}=\frac{(x-3)(x-2)}{(x-2)(x+2)}-\frac{(x+4)(x+2)}{(x-2)(x+2)}
.$$
Now, since the denominators are the same, then we have
$$
\frac{(x-3)(x-2)}{(x-2)(x+2)}-\frac{(x+4)(x+2)}{(x-2)(x+2)}=\frac{x^2-5x+6-(x^2+6x+8)}{(x-2)(x+2)}\\
=\frac{-11x-2}{(x-2)(x+2)}=-\frac{11x+2}{(x-2)(x+2)}
.$$
