Let A be an integral domain, and a,b are elements in A. Let B=A[x]/(ax+b), where (ax+b) is ideal generated by ax+b. Suppose , show that B is an integral domain.

I think it suffices to show that (ax+b) is prime under the assumption, and hence need to show that if f(x)g(x) is in (ax+b), then either f(x) or g(x) is in (ax+b).

thank you for help!