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

[Max Alekseyev <maxale@gmail.com>]

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