Assume it factorises into two polynomials in Q, f=gh. Then f=(1/a)g'h', where a is the product of all the denominators of all the coefficients of g and h, with g' and h' in Z[x]. Then k=g'h' is in Z[x], so f=(1/a)k with k in Z[x]. Can you see where to go from here? The point is that you must be able to `spread out' the integer a between g' and h'.