jmc

To perform division in complex numbers, we multiply both the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate of $1+2i$ is $1-2i$. Multiplying: $$\begin{array}{rl}\frac{-3+4i}{1+2i}& =\frac{(-3+4i)(1-2i)}{(1+2i)(1-2i)}\\ & =\frac{-3+4i+6i-8i^2}{1+2i-2i-4i^2}\\ & =\frac{5+10i}{5}\\ & =\boxed{1+2i}\end{array}$$