### 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 problem involves rational expressions and partial fraction decomposition, which is a technique for writing a single fraction as a sum of simpler fractions.
To find the values of $A$ and $B$, I need to multiply both sides of the equation by the common denominator of $x^2+x-2$, which is $(x+2)(x-1)$.
This gives me $4x+5 = A(x-1) + B(x+2)$.
Now, I can try to find $A$ and $B$ by either equating coefficients of $x$ and constants, or by plugging in convenient values of $x$ that make one of the terms on the right-hand side zero.
I'll use the second method, because it seems easier to me.
If I plug in $x=1$, I get $4(1)+5 = A(1-1) + B(1+2)$, which simplifies to $9=3B$.
Therefore, $B=3$.
If I plug in $x=-2$, I get $4(-2)+5 = A(-2-1) + B(-2+2)$, which simplifies to $-3=-3A$.
Therefore, $A=1$.
Now that I have $A$ and $B$, I can plug them into the expression I need to find: $\frac{B}{x+1} - \frac{A}{x-2}$.
This gives me $\frac{3}{x+1} - \frac{1}{x-2}$.