Write the two term on the left-hand side as one fraction, with denominator (x - 5):
A / (x - 5) + B(x + 1) = A / (x - 5) + B(x + 1)(x - 5) / (x - 5) = [A + B(x + 1)(x - 5)] / (x - 5)
Multiplying out, and simplifying, the numerator:
[A + Bx2 - 4Bx - 5B] / (x - 5) = [ Bx2 - 4Bx +(A - 5B) ] / (x - 5)
Setting the numerator on the left side with the numerator on the right side:
Bx2 - 4Bx +(A - 5B) = -3x2 + 12x + 22
Setting the two x-squared terms equal to each other: Bx2 = -3x2 ---> B = -3
Setting the two x terms equal to each other: -4Bx = 12x ---> -4B = 12 ---> B = -3
Setting the two constant terms equal to each other: A - 5B = 22 ---> A - 5(-3) = 22
---> A + 15 = 22 ---> A = 7