Example 8. If the polynomial $P=2a^{2}-8ab+17b^{2}$ $-16a-4b+1990$, what is the minimum value of $P$? (1990, Junior High School Correspondence Contest in Some Provinces and Cities)
$$
\begin{array}{r}
\text { Sol } P=2 a^{2}-8 a b+17 b^{2}-16 a-4 b \\
+1990=2(a-2 b-4)^{2}+9(b-2)^{2}+1922 .
\end{array}
$$

From this, for all real numbers $a, b, P \geqslant 1922$. The equality holds only when $a-2 b-4=0, b-2=0$, i.e., $a=$ $8, b=2$. Therefore, the minimum value of $P$ is $1922$.