Prove that there are no solutions to $ k^2 = x^4 + 2x^3 + 2x^2 + 2x + 1 $ in $ \mathbb Z^+$.
I have tried a bounding argument so far, placing $k^2$ in between $x^4$ and $(x+1)^4$, but I am unable to prove that this implies there are no solutions.
2026-03-31 12:12:48.1774959168
Impossibility of Equation
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Hint: $$x^4+2x^3+2x^2+2x+1 = (x^2+1)^2 + 2x(x^2+1) = (x^2+1)(x+1)^2.$$