Few years back a question on whether a variant of integer factorization was $\mathsf{NP}$ complete was asked, which according to state of mathematics today does not have a conclusive answer.
Is following variant of polynomial factorization, 'given $p(x)\in\Bbb Z[x], l,u\in\Bbb N$, is there a $q(x)\in\Bbb Z[x]$ with $l\leq\mathsf{deg}(q(x))\leq u$ such that $q(x)|p(x)$', $\mathsf{NP}$ complete?