Example 3 Solve the equation
$$
x^{4}+x^{2}(5 x-3)-2(5 x-3)^{2}=0 \text {. }
$$
Let $5 x-3=y$. Then the original equation becomes
$$
2 y^{2}-x^{2} y-x^{4}=0 \text {. }
$$

Considering it as a quadratic equation in $y$, we solve to get
$$
\begin{array}{l}
y=\frac{x^{2} \pm \sqrt{x^{4}+8 x^{4}}}{4}=\frac{x^{2} \pm 3 x^{2}}{4} \\
\Rightarrow y=x^{2} \text { or } y=-\frac{1}{2} x^{2} \\
\Rightarrow 5 x-3=x^{2} \text { or } 5 x-3=-\frac{1}{2} x^{2} \\
\Rightarrow x=\frac{5 \pm \sqrt{13}}{2},-5 \pm \sqrt{31} .
\end{array}
$$