Example 12. $a, b, c$ are all integers, and $a b c=1990$. Find the minimum value of $a b+b c+c a$. (1990, Jinyun Cup Junior Competition)
$$
\begin{array}{l} 
\text { Sol } \because 1990=1 \times 1 \times 1990 \\
=(-1) \times(-1) \times 1990 \\
= 1 \times(-1) \times(-1990) \\
= 1 \times 2 \times 995 \\
=(-1) \times(-2) \times 995 \\
= \cdots=2 \times 5 \times 199=1 \times 10 \times 199,
\end{array}
$$

When $a=-1, b=-1, c=1990$ (letters can be swapped), $ab+bc+ca=-1990-1990+1=-3979$ is the minimum.
Therefore, the minimum value of $ab+bc+ca$ is -3979.