Problem 577

Let $y=x^{2}+3 x-2$, then $x=y^{2}+3 y-2$.
Subtracting the two equations, we get
$$\begin{aligned} y-x & =\left(x^{2}+3 x-2\right)-\left(y^{2}+3 y-2\right) \\ & =(x-y)(x+y)+3(x-y) . \end{aligned}$$

That is, $(x-y)(x+y+4)=0$,
so $x-y=0$ or $x+y+4=0$.
If $x-y=0$, then $x=y$, substituting into $y=x^{2}+3 x-2$, we get $x$
$$=-1 \pm \sqrt{3} \text {. }$$

If $x+y+4=0$, then $y=-x-4$, substituting into $y=x^{2}+3 x-2$, we get $x=-2 \pm \sqrt{2}$.