Q: Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2},...,f_{n}$ with rational coefficients 
Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2},...,f_{n}$ with rational coefficients satisfying $$x^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+...+f_{n}(x)^{2}.$$ 

It's Olympiad question.
My try is.. The equality $x^{2}+7=x^{2}+2^{2}+1^{2}+1^{2}+1^{2}$,  shows that $n\leq 5$ It remains to show that $x^{2}+7$ is not a sum of four (or less) squares of polynomials with rational coefficients.
is there any one can show me a full solution?
