Learn to use parentheses! This should be

A/(2x+3) + B/(x-4) = (x+1)/ (2x^2 -5x-12)

and

[A(x-4) + B(2x+3)] / [(2x+3)(x-4)] = (x+1)/ [(2x+3)(x-4)]

I haven't checked to make sure your computations are correct, but in terms of general approach you're doing it right, now expand the numerator to

Ax - 4A + 2Bx + 3B = (A + 2B) x + (3B - 4A)

Set up system of linear equations

A + 2B = 1

3B - 4A = 1

Now solve for A and B.