<s>[INST] Greatest common divisor of $(x+1)^{4n+3} + x^{2n}$ and $x^3-1$. I have to find the greatest common divisor of 
$$(x+1)^{4n+3} + x^{2n}$$
and
$$x^3-1$$
I know I can express the second polynomial as:
$$x^3-1 = (x-1)(x^2+x+1)$$
So I would have to check if the first polynomial is divisible by $(x^3-1)$, $(x^2+x+1)$ or $(x-1)$ and if it is not divisible by any of those, then the two polynomials do not have a common divisor except for $1$. But I don't know how I can divide the polynomial
$$(x+1)^{4n+3} + x^{2n}$$
by those $3$ other polynomials and therefore can't check the greatest common divisor.
 [/INST] Only $x^2+x+1$ can be an answer because $x-1$ is not valid.
Modulo $x^2+x+1$ we obtain: 
$$(x+1)^{4n+3}+x^{2n}\equiv x^{2n}-x^{8n+6}=x^{2n}(1-(x^3)^{2n+2})\equiv0.$$
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