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.

Let $Q(x) = (x+1)^{4n+3} + x^{2n}$. Then $Q(1) = 2^{4n+3} + 1 \neq 0$, hence $x-1$ does not divide $Q(x)$. 
We have $$Q(x) = (x+1)^{4n+3} + x^{2n} = (x+1)[(x+1)^2]^{2n+1} + x^{2n} =$$
$$= (x+1)[(x^2+x+1) + x]^{2n+1}+ x^{2n} = $$
$$=(x^2+x+1)P(x) + (x+1)x^{2n+1} + x^{2n} = (x^2+x+1)(P(x) + x^{2n}),$$
hence $x^2+x+1$ divides $Q(x)$.
