### 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: To solve this problem, I need to find the values of $A$ and $B$ that make the equation true for any value of $x$.
One way to do that is to multiply both sides of the equation by the common denominator $(x+2)(x-1)$ and simplify.
That gives me $$4x+5 = A(x-1) + B(x+2).$$
Now I have a linear equation in $x$ that must be true for all $x$, so the coefficients of $x$ on both sides must be equal, and the constant terms must be equal.
That means I have two equations: $$4 = A + B$$ and $$5 = -A + 2B.$$
I can solve this system of equations by adding them together and dividing by $3$: $$9 = 3B$$ so $$B = 3.$$
Then I can plug in $B = 3$ into either equation to find $A$: $$4 = A + 3$$ so $$A = 1.$$
Now that I have $A$ and $B$, I can plug them into the expression I want to find: $$\frac{B}{x+1} - \frac{A}{x-2} = \frac{3}{x+1} - \frac{1}{x-2}.$$