Example 4 Solve the equation $y^{4}-6 y^{3}+10 y^{2}-4 y=0$.

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Example 4 Solve the equation $y^{4}-6 y^{3}+10 y^{2}-4 y=0$.
The original equation can be transformed into $\left(y^{2}-3 y+2\right)^{2}-3\left(y^{2}-3 y+2\right)+2-y=0$. Let $f(y)=$ $y^{2}-3 y+2$, then the two fixed points of $f(y)$ are $2+\sqrt{2}$ and $2-\sqrt{2}$. Therefore, $2+\sqrt{2}$ and $2-\sqrt{2}$ are also two fixed points of $f[f(y)]$. The other two fixed points of $f[f(y)]$ are obtained from $1+[1+(y-2+\sqrt{2})][1+(y-2-\sqrt{2})]=0$, which simplifies to $y^{2}-2 y=0$, yielding 0 and 2. Hence, the four roots of the equation are $2+\sqrt{2}$, $2-\sqrt{2}$, 0, and 2.