Algebra Review for Pre-Calculus (follow-up)

I recently wrote about Algebra Review for Pre-Calculus and discussed some problems that I typically assign in the College Algebra courses I teach. Here is a link to some examples of those problems including algebraic simplification, simplifying rational expressions and solving equations involving rational expressions.

I'll write next about the Rational Roots Theorem and polynomial factorization. Here's a problem I found from a Japanese University Entrance Exam from 1990 that makes interesting use of these concepts. The document I took this from is here - the problem is from the first sample test on page 4.

Suppose the polynomial P(x) with integer coefficients satisfies the following conditions:

(A) If P(x) is divided by x^2-4x+3, the remainder is 65x-68

(B) If P(x) is divided by x^2+6x-7, the remainder is -5x+a

Then we know that a={?}.

Let us find the remainder bx+c when P(x) is divided by x^2+4x-21.
Condition (A) implies that {?}b+c={?}.
Condition (B) implies that {?}b+c={?}.
It follows that b={?} and c={?}

I've changed the notation of the answers a little in the hope of making it less confusing - you can see the original by clicking through the link.

I got pretty tangled up in solving this problem because I had never seen the Remainder Theorem used in quite this way before. The answers for this are on page 5 of the original.