Example 2 Calculate $(1+2 i) \div(3-4 i)$.

. 

The translation maintains the original text's line breaks and format, as requested.
\[
\begin{array}{l}
(1+2 i) \div(3-4 i)=\frac{1+2 i}{3-4 i}=\frac{(1+2 i)(3+4 i)}{(3-4 i)(3+4 i)} \\
=\frac{3+4 i+6 i+8 i^{2}}{9+12 i-12 i-16 i^{2}}=\frac{(3-8)+(4+6) i}{9+16} \\
=\frac{-5+10 i}{25}=\frac{-1+2 i}{5}
\end{array}
\]