I'd say that this is a case of the chinese remainder theorem for a polynomial ring.

You're told that and , for some polynomials q(x), r(x). Multiply the first of those equations by x–6 and the second one by x+3, and subtract. You'll get , for some polynomial s(x). Divide by –9 to find the remainder when p(x) is divided by (x+3)(x-6).