Let $f_1,f_2,f_3,f_4,f_5 \in \mathbb{Q}[x]$ be linear and coprime and not all constant.

Is it possible $ f_1^2+f_2^2+f_3^2+f_4^2=f_5^2$?

I suppose the answer is negative.

If this is possible, solving $f_5(x)=N$ would give deterministic representation of $N$ as sum of four squares (probabilistic algorithms exist).

Couldn't solve this by equating coefficients.

What is the smallest natural $k$ such that subject to the same constraints $f_1^2+f_2^2+\cdots+f_k^2=f_{k+1}^2$?