We are to assume here that f(x), for integer x is integer valued? That is the same as saying the coefficients are integers.

"Congruent as polynomials modulo n" means that for every integer x, f1(x)= g1(x)+ kn for some integer k. Similarly, f2(x)= g2(x)+ hn.

Then f1(x)+ f2(x)= g1(x)+ kn+ g2(x)+ hn= (g1(x)+ g2(x))+ (k+ h)n.

f1(x)f2(x)= (g1(x)+ kn)(g2(x)+ hn)= g1(x)g2(x)+ hng1(x)+ kng2(x)+ hkn^2= g1(x)g2(x)+ (hg1(x)+ kg2(x)+ khn)n.