I will assume that the degree of the polynomial is the number of complex roots.And all the polynomials mentioned have finite degree.
1) Count the number of roots. for p(x)q(x) = 0, roots of any of the polynomial p(x) or q(x) will do. So there are totally deg(p(x)) + deg (q(x)) roots. Thus the answer.
2)If for all values of x, we have r(x).p(x) = r(x).q(x), then for all values of x, r(x).[p(x)-q(x)] = 0. This forces [p(x)-q(x)] to be identically zero (since r(x) is non zero). Thus p(x) = q(x).
P.S: I think TPH and Mathguru had a similar discussion in the forum somewhere