262. Integer solutions are sought. Find all integer solutions of the equation

$$
y^{2}+y=x^{4}+x^{3}+x^{2}+x
$$
262. Multiplying the given equation by 4 and then adding 1 to both sides, we get

$$
4 x^{4}+4 x^{3}+4 x^{2}+4 x+1=(2 y+1)^{2}
$$

For $x=-1$, $y=-1$ or 0; for $x=0$, $y=-1$ or 0; for $x=2$, $y=-6$ or 5; for $x=1$, $y$ will not be an integer.

These are the only 6 integer solutions of the given equation. Indeed, for $x>2$, the left side of the transformed equation will be greater than $\left(2 x^{2}+x\right)^{2}$ but less than $\left(2 x^{2}+x+1\right)^{2}$. Therefore, for no integer $x$ in these ranges can the left side equal a perfect square.

[D. Marsh, A. M. M., 73, 895 (October 1966).]