### Instruction: $A$ and $B$ are constants such that
$$\frac{4x+5}{x^2+x-2}= \frac{A}{x+2} +\frac{B}{x-1}. $$Find $$\frac{B}{x+1} - \frac{A}{x-2}.$$.### Response: This looks like a problem involving partial fraction decomposition, where we want to write a rational function as a sum of simpler fractions.
To do that, we need to find the values of $A$ and $B$ that make the equation true for all $x \neq -2, 1$, since those are the values that make the denominators zero.
One way to find $A$ and $B$ is to multiply both sides of the equation by the common denominator $(x^2+x-2) = (x+2)(x-1)$ and then simplify.
Doing that, we get 
$$4x+5 = A(x-1) + B(x+2).$$
Now we can compare the coefficients of the different powers of $x$ on both sides.
For the constant term, we have $5 = -A + 2B$, or $A = 2B - 5$.
For the linear term, we have $4 = A + B$, or $B = 4 - A$.
Substituting one equation into the other, we get $A = 2(4-A) - 5$, or $3A = 3$, or $A = 1$.
Then we can plug in $A = 1$ into either equation to get $B = 3$.
So we have 
$$\frac{4x+5}{x^2+x-2}= \frac{1}{x+2} +\frac{3}{x-1}.
$$
Now we can find the expression we want by substituting these values of $A$ and $B$ and simplifying.
We have 
\begin{align*}
\frac{B}{x+1} - \frac{A}{x-2} &= \frac{3}{x+1} - \frac{1}{x-2} \\
&= \frac{3(x-2) - (x+1)}{(x+1)(x-2)} \\
&= \frac{2x-7}{x^2-x-2}.
\end{align*}