jmc

We notice that we can get an expression like $A-B$ by substituting $x=0$. If $x=0$ we get $$\frac{A}{0-1}+\frac{B}{0+1}=\frac{0+2}{0^2-1},$$or $$-A+B=-2.$$Multiplying by $-1$ gives $$A-B=\boxed{2}.$$