Factorise $x^{111}+9x^{74}+27x^{37}+27$ in $\mathbb{Z}[x]$

Question

I am having trouble with the following.

Factorise $x^{111}+9x^{74}+27x^{37}+27$ in irreducible factors in $\mathbb{Z}[x]$.

I did not find it to be an Eisentein polynomial and trying to find zeros by hand is a lot of work, so is a linear substitution of $x$ by $cx+b$. Also, reducing the polynomial modulo a prime number (for example 3) did not help.

Answer 1

Hint:

Make the substitution $u=x^{37}$

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For completion. After the hint, you can write your expression like $$\color{green}{1}\cdot 3^0\cdot u^3+\color{green}{3}\cdot 3^1\cdot u^2+\color{green}{3}\cdot 3^2\cdot u+\color{green}{1}\cdot 3^3\\(u+3)^3=(x^{37}+3)^3$$