Nice approach! I would try plugging in specific values of $x$. The $A,B,C$ relation you have found must hold for all $x$, so it should hold for, say, $x=1$ and $x=-3/2$. Then I would just pick another nice value for $x$, say, $x=0$. You get three equations in three unknowns, technically, but I don't think it'll end up being that complicated.