jmc

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 $$\begin{array}{rl}\frac{B}{x+1}-\frac{A}{x-2}& =\frac{3}{x+1}-\frac{1}{x-2}\\ & =\frac{3(x-2)-1(x+1)}{(x+1)(x-2)}\\ & =\boxed{\frac{2x-7}{(x+1)(x-2)}}.\end{array}$$