8. Find the integer solutions of the equation $x^{2}+x=y^{4}+y^{3}+y^{2}+y$.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
(Hint: Transform the original equation into
$$
(2 x+1)^{2}=\left(2 y^{2}+y+1\right)^{2}+2 y-y^{2} \text {. }
$$

If $(3 y+1)(y+1)>0$ and $2 y-y^{2}<0$, then it contradicts that $(2 x+1)^{2}$ is a perfect square. Therefore,
$$
(3 y+1)(y+1) \leqslant 0 \text { or } 2 y-y^{2} \geqslant 0 \text {, }
$$

which means $-1 \leqslant y \leqslant-\frac{1}{3}$ or $0 \leqslant y \leqslant 2$.
Thus, $y=-1,0,1,2$.
$$
\begin{aligned}
(x, y)= & (0,-1),(-1,-1),(0,0), \\
& (-1,0),(5,2),(-6,2) .
\end{aligned}
$$