jmc

First we try factoring the left side to simplify it: $$\frac{4x-12}{x^2+2x-15}=\frac{4(x-3)}{(x-3)(x+5)}=\frac{4}{x+5}.$$Now we can multiply both sides by $(x+5)$ and solve for $x$: $$\begin{array}{rl}\frac{4}{x+5}& =x+2\Rightarrow \\ 4& =(x+5)(x+2)\Rightarrow \\ & =x^2+7x+10\Rightarrow \\ 0& =x^2+7x+6\Rightarrow \\ & =(x+6)(x+1).\end{array}$$So $p=-1$ and $q=-6$, making $p-q=\boxed{5}$.