MathNet 2019-part-07 — 184 problems (EN–VI) Question 173

Let $P(x)$ be a polynomial with real coefficients so that $P(x) \geq 0$ for all real $x$. Prove that there exist polynomials $Q_{1}(x)$ and $Q_{2}(x)$ with real coefficients such that $P(x) = Q_{1}^{2}(x) + Q_{2}^{2}(x)$ for all $x$.