Let $\mathbb F_q$ be a finite field and $g(x)$ be a polynomial in $\mathbb F_q[x]$ with non zero constant term. Given the factorization of $g(x)=p_1(x)^{e_1}\dots p_k(x)^{e_k}$, give an estimate for the order of $x$ in $(\mathbb F_q[x]/(g(x)))^\times$.
Using Chinese Remainder Theorem and basic properties of the order and lcm, one gets that the order of $x$ divides $$ C(g)=q^d(q-1) \cdot \operatorname{lcm}(\{(q^{\deg p_i}-1)/(q-1)\}_{i\in \{1\dots k\}}), $$ where $d$ is the smallest integer such that $q^d$ is larger than any of the $e_i$'s. But I am not really able to nicely estimate this quantity...