Example 5. Solve the equation:
$$
5 x^{2}-6 x y+2 y^{2}-4 x+2 y+1=0 \text {. }
$$
Solution: The original equation can be transformed into
$$
\begin{array}{l}
\left(x^{2}-2 x y+y^{2}\right)+\left(4 x^{2}-4 x y\right. \\
\left.+y^{2}\right)-4 x+2 y+1=0, \\
(x-y)^{2}+(2 x-y)^{2} \\
-2(2 x-y)+1=0, \\
(x-y)^{2}+(2 x-y-1)^{2}=0 .
\end{array}
$$

Using the method from Example 4, we get
$$
\left\{\begin{array}{l}
x=1, \\
y=1 .
\end{array}\right. \text { as the solution to the original equation. }
$$