How to prove that $4x^3+6x^2+4x+1$ is not a fourth power of an integer, for any $x\in\mathbb N$? 
How do I prove that for all positive integers $x$ it's true that, $4x^3+6x^2+4x+1$ is not a fourth power of an integer?

I've tried doing modulo 3 and 5 checks, and it didn't really go far from there

If $4x^3+6x^2+4x+1=w^4$, then:
$$ x^4 + w^4 = (x+1)^4 $$
but such an identity contradicts the case $n=4$ of the Fermat's last theorem (there is an elementary proof of that case in terms of Gaussian integers, so the full power of the general FLT is not needed).
