MHB Prove y is not a perfect square

[Albert1]

$x\in N$
$y=x^4+2x^3+2x^2+2x+1$
prove:$y$ is not a perfect square

 

[mathbalarka]

Spoiler

For all $x > 0$, $$(x^2 + x)^2 = x^4 + 2x^3 + x^2 < x^4 + 2x^3 + 2x^2 + 2x + 1 < x^4 + 2x^3 + 3x^2 + 2x + 1 = (x^2 + x + 1)^2$$ As $y$ is sitting in between two consecutive perfect squares for $x \in \Bbb N \setminus \{0\}$, $y$ cannot itself be a perfect square.

 

[kaliprasad]

Spoiler

$y=x^4+2x^3+2x^2+2x+1$
=$x^4+2x^2+1+2x^3+2x$
=$(x^2+1)^2+2x(x^2+1)$
= $(x^2+1)(x^2+2x+1)$
= $(x^2+1)(x+1)^2$
as $x^2+1$ is between $x^2$ and $(x+1)^2$ and not a perfect square and $(x+1)^2$ is so the product is not a perfect square