Find the $gcd(x^8, x^6+x^4+x^2+x+1)$ using the euclidean algorithm.
$x^8 = (x^2)(x^6+x^4+x^2+x+1)+(-x^6-x^4-x^3-x^2)$ $(x^6+x^4+x^2+x+1)=(-1)(-x^6-x^4-x^3-x^2)+(-x^3+x+1)$ $(-x^6-x^4-x^3-x^2)=(x^3)(-x^3+x+1)+(-2x^4-2x^3-x^2)$
I'm not quite too sure where to go from here because $deg(-2x^4-2x^3-x^2) >deg(-x^3+x+1)$. Did I factor incorrectly?
Rewrite your work as:
$$\begin{align} \gcd(x^8, x^6 + x^4 + x^2 + x + 1) & = \gcd(x^6 + x^4 + x^2 + x + 1, -x^6 - x^4 - x^3 - x^2)\\ & = \gcd(-x^6 - x^4 - x^3 - x^2, -x^3 + x + 1)\\ & = \gcd(-x^3 + x + 1, -2x^4 - 2 x^3 - x^2)\\ \end{align} $$
Since $\gcd(a,b) = \gcd(b,a)$, you're fine. Just keep at it.