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).
Note we can write:
y⁴ = 4x³ + 6x² + 4x + 1
as: 
y⁴ = (x + 1)⁴ - x⁴
and as mentioned is a form of Fermat's Last Theorem.
→ x⁴ + y⁴ = (x + 1)⁴
This instance of FLT was put to rest long before Wiles finally put the problem to rest.  Check out this simple proof:
http://fermatslasttheorem.blogspot.com/2005/05/fermats-last-theorem-n-4.html?m=1