Example 3: If $a^{2}+b^{2}+c^{2}+3 \leqslant a b+3 b+2 c$, find the values of $a$, $b$, and $c$.
(1992, Hungarian Mathematical Olympiad)
Solution: The original inequality is equivalent to
$$
\left(a-\frac{b}{2}\right)^{2}+\frac{3}{4}(b-2)^{2}+(c-1)^{2} \leqslant 0 .
$$

By the property of non-negative numbers, we have
$$
a-\frac{b}{2}=b-2=c-1=0 \text {. }
$$

Solving, we get $a=1, b=2, c=1$.