Solve $x^2 + x = y^4 + y^3 + y^2 + y$ over integers.

Question

I am trying to solve solve $x^2 + x = y^4 + y^3 + y^2 + y$ over the integers. So far I have decomposed it into $x(x + 1) = y(y + 1)(y^2 + 1)$ and noticed that both sides of the equation are nonnegative. Furthermore $GCD(x, x + 1) = GCD(y, y + 1) = GCD(y, y^2 + 1) = GCD(y + 1, y^2 + 1) = 1$.

Answer 1

Hint: If you multiply both sides by $4$ and add $1$, you get

$$(2x+1)^2=4y^4+4y^3+4y^2+4y+1=(2y^2+y+1)^2-(y^2-2y).$$

This results in two squares that are very close together. Can you prove that they are too close together when $y$ is sufficiently large?