Example 1. Let the integer part of $\frac{1}{3-\sqrt{7}}$ be $a$, and the decimal part be $b$, find the value of $a^{2}+(1+\sqrt{7}) a b$.
$$
\begin{aligned}
\because \frac{1}{3-\sqrt{7}}=\frac{3+\sqrt{7}}{(3-\sqrt{7})(3+\sqrt{7})} \\
\quad=\frac{3+\sqrt{7}}{2},
\end{aligned}
$$

and $2<\sqrt{7}<3,0<\frac{\sqrt{7}-1}{2}<1$,
$$
\begin{array}{l} 
\therefore \quad \frac{1}{3-\sqrt{7}}=\frac{3+1+\sqrt{7}-1}{2} \\
=2+\frac{\sqrt{7}-1}{2} .
\end{array}
$$

Therefore, $a=2, b=\frac{\sqrt{7}-1}{2}$,
$$
\begin{array}{l}
\text { Hence } \mathrm{a}^{2}+(1+\sqrt{7}) \mathrm{ab}=2^{2} \\
+\frac{2(1+\sqrt{7})(\sqrt{7}-1)}{2}=4+6=10 .
\end{array}
$$