Question 14
Let $P(x)$, $Q(x)$ be polynomials with real coefficients such that $P(0) \gt 0$ and all coefficients of the polynomial $S(x) = P(x)Q(x)$ are non-negative. Prove that for any positive $x$ the following inequality holds:
$$
S(x^2) - S^2(x) \le \frac{1}{4}(P^2(x^3) + Q(x^3)).
$$
$$
S(x^2) - S^2(x) \le \frac{1}{4}(P^2(x^3) + Q(x^3)).
$$