Example 13. Find the integer solutions of the equation $y^{2}+y=x^{4}+x^{8}+x^{2}+x$.
The original equation is transformed into
$$
(2 y+1)^{2}=4 x^{4}+4 x^{3}+4 x^{2}+4 x+1 \text {. }
$$

Taking $2 x^{2}+x$ and $2 x^{2}+x+1$, for all integer $x$ we have $\left(2 x^{2}+x\right)^{2}0, \\ x^{2}-2 x>0 .\end{array}\right.$
Therefore, there is no solution within $x2$, which should be eliminated.
Thus, $x=-2,-1,0,1$. Finally, the solutions are
$$
\left\{\begin{array}{l}
x=-1,-1,0,0,2,2, \\
y=-1,0,-1,0,-6,5
\end{array}\right.
$$