For what $n$ is $x^2 + x+ 1\mid x^{2n} + x^n + 1$? For what $n \in \mathbb{N}$ is $x^2 + x+ 1\mid x^{2n} + x^n + 1$?
The only obvious thing that I could see was noticing that $(x^3 - 1) = (x-1)(x^2 + x+1)$. So, if $x^3 - 1\mid x^{2n} + x^n + 1$. I don't even know if this will help. 

HINT:
If $\omega$ is a root of $x^2+x+1=0, \omega^2+\omega+1=0\implies \omega^3=1$
So if $3|n, n=3m$(say),
$$\omega^n=\omega^{3m}=(\omega^3)^m=1 \text{ and } \omega^{2n}=\omega^{6m}=(\omega^3)^{2m}=1$$
$$\implies \omega^{2n}+\omega^n+1=3\ne0$$
$\implies \omega$ is not a root of $x^{2n}+x^n+1=0$ 
If $n=3m+1,\omega^{2n}+\omega^n+1=\omega^{2(3m+1)}+\omega^{3m+1}+1=\omega^2+\omega+1=0$
$\implies \omega$ is a root of $x^{2n}+x^n+1=0$ 
$\implies (x-\omega)|(x^{2n}+x^n+1)$
Similarly, $(x-\omega^2)|(x^{2n}+x^n+1)$
and as $(x-\omega)(x-\omega^2)=x^2+x+1\implies (x^2+x+1)|(x^{2n}+x^n+1)$
Similarly, for $n=3m+2$
