find the inverse of $x^2 + x + 1$ In $\mathbb{F}_2[x]$ modulo $x^4 + x + 1$
find the inverse of $x^2 + x + 1$
not 100% sure but here what i have:
user euclid algorithm:
$x^4 + x + 1 = (x^3 + 1)(x + 1) + x$
$(x^3 + 1) = x * x * x + 1$
$1 = (x^3 + 1) - x * x * x  $

Using that $\;x^4=x+1\; $ in $\,\Bbb F_2[x]/(x^4+x+1)\;$ , prove that
$$x^2+x=(x^2+x+1)^{-1}\;\;\; (\text{ further hint:}\;(x^2+x+1)^3=1)$$
