Ascending chain of ideal in polynomial ring

Question

I have an ideal $I_{1}=\langle x^5+x\rangle$, I have to find the largest $n$ so that $I_{1} \subset I_{2} \subset ... \subset I_{n} \subset \mathbb{Q}[x] $. I was thinking since $a|b \iff \langle b\rangle \subset \langle a\rangle$ and $x^5-x=x(x^4-1)=x(x^2+1)(x+1)(x-1)$ so there are four divisors of this polymonial, hence $n=4$. I'm not sure though if my reasoning is fine.