$f(x)=2x^{3}+4x^{2}+2x+2$ and $g(x)=2x^{2}+x+3$ in $\mathbb{Z}_5[x]$ $$I=<f(x)>$$
I want to know if there is such natutal number $n$, and if there is to find, otherwise to prove that there isn't: $$(g(x)+I)^{n}=I$$
Can I solve this without using euclidean algorithm for polynomials ? I do know that $f(x)=(x+4)(2x^{2}+x+3)$ then $(f,g)=2x^{2}+x+3$