finding primes modulo which x^m mod f(x) has a prescribed result

Hello,

Suppose I have a large number m, a quadratic polynomial f(x) and linear
polynomial g(x).
Is there a fast way to find all primes p such that the remainder of
division of (x^m - g(x)) by f(x) vanishes modulo p ?
To give a specific example, let m = 10^10, f(x) = x^2 - 3*x - 3, and g(x) =
x - 4.

Thanks,
Max