Find the partial fraction decomposition of the expression: (4x^2 + x -64)/((x+2)(x-3)(x-4)).

The first step to solving this question is inputting our dummy variables, and laying out the question so we know what we're doing.
So we end up with:(4x² + x - 64)/((x+2)(x-3)(x-4)) = A/(x+2) + B/(x-3) + C/(x-4).
The next step is to clean up this equation and make it a bit easier to substitute values into it. This is done by multiplying each side through by the denominator.
4x² + x - 64 = A(x-4)(x-3) + B(x+2)(x-4) + C(x+2)(x-3).
We know that this expression needs to hold for every value of x. This means that we can cherry pick values of x to try and gain information about our unknowns.
The first case is x = -2.
So we substitute x = -2 in and end up with:
4(-2)² - 2 - 64 = A(-2-4)(-2-3) + B(-2+2)(-2-4) + C(-2+2)(-2-3).
Choosing x = -2 as the value to look at has the advantage of turning any (x+2) factors into a 0. This means that both B and C now have 0 as their coefficient. This means that A will be our only variable, and we can solve for it easily.
16 - 2 - 64 = A(-6)(-5).
-50 = 30A.
A = -5/3.
Our next two cases are x = 3 and x = 4. These are solved in a similar way to the x = -2 case.
Once substituted into the expression x = 3 gives a value of B = 5, and x = 4 gives a value of C = 2/3.The last step is to substitute these back into our original expression, which gives an answer of:
(4x² + x - 64)/((x+2)(x-3)(x-4)) = -5/3(x+2) + 5/(x-3) + 2/3(x-4).