2. If the polynomial $P=2 a^{2}-8 a b+17 b^{2}-16 a-4 b$ +2070, then the minimum value of $P$ is
$$
\begin{array}{l}
2.2002 \\
P=2 a^{2}-8(b+2) a+17 b^{2}-4 b+2070 \\
=2\left[a^{2}-4(b+2) a+4(b+2)^{2}\right]-8(b+2)^{2} \\
\quad+17 b^{2}-4 b+2070 \\
=2(a-2 b-4)^{2}+9 b^{2}-36 b+36+2002 \\
=2(a-2 b-4)^{2}+9(b-2)^{2}+2002
\end{array}
$$

When $a-2 b-4=0$ and $b-2=0$, $P$ reaches its minimum value of 2002.