Example 1 Find integers $a, b, c$, such that they satisfy the condition:
$$
a^{2}+b^{2}+c^{2}+3 < a b + 3 b + 2 c \text{. }
$$
Given the condition $a^{2}+b^{2}+c^{2}+3 < a b+3 b+2 c$ and the discreteness of integers, we have $a^{2}+b^{2}+c^{2}+4 \leqslant a b+3 b+2 c$,
transforming the above equation by completing the square, we get $\left(a-\frac{b}{2}\right)^{2}+3\left(\frac{b}{2}-1\right)^{2}+(c-1)^{2} \leqslant 0$.
Since $\left(a-\frac{b}{2}\right)^{2} \geqslant 0,\left(\frac{b}{2}-1\right)^{2} \geqslant 0,(c-1)^{2} \geqslant 0$,
it follows that $a-\frac{b}{2}=\frac{b}{2}-1=c-1=0$.
Therefore, $a=1, b=2, c=1$ is the solution.