Arithmetic properties of polynomials

First, we prove that the Diophantine system f (z) = f (x) + f (y) = f (u) - f (v) = f (p) f (q) has infinitely many integer solutions for f (X) = X(X + a) with nonzero integers a = 0, 1, 4 (mod 5). Second, we show that the above Diophantine system has an integer parametric solution for f (X) = X(X + a) with nonzero integers a, if there are integers m, n, k such that (n2 - m2)(4mnk(k + a + 1) + a(m2 + 2mn - n2)) = 0 (mod (m2 + n2)2), (m2 + 2mn - n2)((m2 - 2mn - n2)k(k + a + 1) - 2amn) = 0 (mod (m2 + n2)2), where k = 0 (mod 4) when a is even, and k = 2 (mod 4) when a is odd. Third, we get that the Diophantine system f (z) = f (x) + f (y) = f (u) - f (v) = f (p) f (q) = f (r) f (s) has a five-parameter rational solution for f (X) = X(X + a) with nonzero rational number a and infinitely many nontrivial rational parametric solutions for f (X) = X(X + a)(X + b) with nonzero integers a, b and a = b. Finally, we raise some related questions.