I know that, in $\Bbb Z$, every nonnegative number can be expressed as the sum of four squares (by Lagrange's four-square identity), and so every nonnegative number can also be expressed as the sum of five squares (by letting the fifth square be $0^2$). Thus, the product of two sums of five squares is also a sum of five squares.
Is this still true in $\Bbb Z[x]$? That is, if two polynomials are each the sum of five squares, is their product also the sum of five squares?
I know that this is true for four squares, using Euler's four-square identity. I also know it's not trivial, since not every element of $\Bbb Z[x]$ is the sum of five squares ($x$ itself being an example). However, I have no idea how I'd approach this problem, mostly because checking whether any given polynomial is the sum of five squares seems to be a very hard computation.