3. Let $M=\frac{8}{\sqrt{2008}-44}$, $a$ is the integer part of $M$, and $b$ is the fractional part of $M$. Then
$$
a^{2}+3(\sqrt{2008}+37) a b+10=
$$
3.2008 .

Since $M=\frac{8}{\sqrt{2008}-44}=\frac{\sqrt{2008}+44}{9}$
$$
\begin{array}{l}
=9+\frac{\sqrt{2008}-37}{9}, \\
0<\frac{\sqrt{2008}-37}{9}<1,
\end{array}
$$

Therefore, $a=9, b=\frac{\sqrt{2008}-37}{9}$.
Then $a b=\sqrt{2008}-37$.
Hence $a^{2}+3(\sqrt{2008}+37) a b+10$
$$
\begin{array}{l}
=9^{2}+3(\sqrt{2008}+37)(\sqrt{2008}-37)+10 \\
=2008
\end{array}
$$