Splitting sums of binary polynomials

We study an analogue of a classical arithmetic problem over the ring of polynomials. We prove that $m=5$ is the minimal number such that the sums of any two distinct polynomials in a set of $m$ polynomials over ${\text{F}}_2[x]$ cannot all be of the form $x^k(x+1{)}^{\mathrm{\ell }}$.