We have that $$\frac{4x+5}{x^2+x-2}= \frac{4x+5}{(x+1)(x-2)}=\frac{A}{x+2} +\frac{B}{x-1}.$$Multiplying both sides by $ (x+1)(x-2) $ gives $$4x+5=A(x+1)+B(x-2)$$which we can expand to get $$4x+5=(A+B)x+A-2B.$$By comparing coefficients, we see that $ A+B=4 $ and $ A-2B=5 $. We can solve these equations to get $ A=1 $ and $ B=3 $. Then $$\begin{aligned} \frac{B}{x+1} - \frac{A}{x-2}&=\frac{3}{x+1} - \frac{1}{x-2}\\ &=\frac{3(x-2)-1(x+1)}{(x+1)(x-2)}\\ &=\boxed{\frac{2x-7}{(x+1)(x-2)}}.\end{aligned}$$