jmc

We want to break up the rational function on the right into a polynomial and a term with constant numerator. To do this, we notice that $-3x^2+15x$ is a multiple of $x-5$, hence $$\frac{-3x^2+12x+22}{x-5}=\frac{-3x^2+15x-15x+12x+22}{x-5}=-3x+\frac{-3x+22}{x-5}.$$Now notice that $-3x+15$ is also a multiple of $x-5$, so $$-3x+\frac{-3x+22}{x-5}=-3x+\frac{-3x+15+7}{x-5}=-3x-3+\frac{7}{x-5}.$$Thus $B=-3$ and $A=7$, so $A+B=\boxed{4}$.