### 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: I want to find the values of $A$ and $B$ that make the equation true for any value of $x$.
To do that, I can multiply both sides of the equation by the common denominator $x^2+x-2$ and simplify.
This gives me $4x+5 = A(x-1) + B(x+2)$.
Now I can compare the coefficients of $x$ and the constant terms on both sides.
For the coefficients of $x$, I get $4 = A + B$.
For the constant terms, I get $5 = -A + 2B$.
I can solve this system of equations by adding the two equations together.
This eliminates $A$ and gives me $9 = 3B$, so $B = 3$.
Then I can substitute $B = 3$ into either equation to find $A$.
For example, using the first equation, I get $4 = A + 3$, so $A = 1$.
Now I have the values of $A$ and $B$, so I can plug them into the expression I want to find.
This gives me $\frac{B}{x+1} - \frac{A}{x-2} = \frac{3}{x+1} - \frac{1}{x-2}$.