jmc

We solve for $A$, $B$, and $C$ using the features of the graph.

We see that the graph passes through the point $(4,0)$, which gives us the equation $$\frac{4+A}{4B+C}=0.$$Therefore, $A=-4$.

We see that the graph passes through the point $(0,-2)$, which gives us the equation $$\frac{0-4}{C}=-2.$$Therefore, $C=2$.

Finally, we see that the graph passes through the point $(3,1)$, which gives us the equation $$\frac{3-4}{3B+2}=1.$$Solving for $B$, we find $B=-1$.

Hence, $A+B+C=(-4)+2+(-1)=\boxed{-3}$.