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

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
