Let $F$ be a finite field of characteristic $2$. Let $m \geq 2 $ be a positive integer.
Seems unknown
if the trinomial
$$
T(t,x) = x^{2m+1}+x^2+s(t) \in F[t][x]
$$
(more explicitly, the constant coefficient $s(t)$ is a polynomial in $t$,
i.e., $s(t) \in F[t]$)
has monic factors $D(t,x)$ in $F[t][x]$ with degree $3$ relative to $x$.
I.e., $$ D(t,x) = x^3+ a_2(t)x^2+a_1(t)x + a_0(t) $$ with $a_2(t), a_1(t),a_0(t) \in F[t]$ and $T(t,x) = D(t,x)K(t,x)$ for some $K(t,x) \in F[t][x].$
We will then say that $T(t,x)$ have $D(t,x)$ as a factor or that $D(t,x)$ divides $T(t,x)$.
Question: Assume that $D(t,x)$ as above divides the trinomial $T(t,x).$
Do we have $$ \deg(a_1(t)) = 2 \deg(a_2(t)),\;\;\deg(a_0(t))=3\deg(a_2(t)). $$ ???
I am aware of the work of Schinzel on trinomials. The trinomials $T(t,x)$ do not seem to be worked out in these papers.
