jmc

We can approach this problem by picking clever values for $x$. If $x=-2$ we get $$\frac{A}{-2-1}+\frac{B}{-2+1}=0,$$ so $$A+3B=0.$$

If $x=0$ we get $$\frac{A}{0-1}+\frac{B}{0+1}=\frac{0+2}{0^2-1},$$ or $$-A+B=-2.$$ To solve for $B$ we add these two expressions: $$4B=-2,$$ so $B=\boxed{-\frac{1}{2}}$.