Prove that a positive polynomial function can be written as the squares of two polynomial functions 
Let $f(x)$ be a polynomial function with real coefficients such that $f(x)\geq 0 \;\forall x\in\Bbb R$. Prove that there exist polynomials $A(x),B(x)$ with real coeficients such that $f(x)=A^2(x)+B^2(x)\;\forall x\in\Bbb R$

I don't know how to approach this, apart from some cases of specific polynomials that turned out really ugly. Any hints to point me to the right direction?

Survey article by Bruce Reznick called Some Concrete Aspects of Hilbert's 17th Problem, includes your case in the paragraph on Before 1900:

