$\begin{array}{l}
4 x^{2}-2\left(a^{2}+b^{2}\right) x+a^{2} b^{2}=0 \\
\Rightarrow 4 x^{2}-2 a^{2} x-2 b^{2} x+a^{2} b^{2}=0 \\
\Rightarrow 2 x\left(2 x-a^{2}\right)-b^{2}\left(2 x-a^{2}\right)=0 \\
\Rightarrow\left(2 x-b^{2}\right)\left(2 x-a^{2}\right)=0 \\
\Rightarrow 2 x-b^{2}=0 \text { or } 2 x-a^{2}=0 \\
\Rightarrow x=\frac{b^{2}}{2} \text { or } x=\frac{a^{2}}{2}
\end{array}$

Hence, the roots of the equation are $\frac{b^{2}}{2}$ and $\frac{a^{2}}{2}$.