Dividing Complex Numbers

Example:

Divide $3+5i$ by $-1+2i$ .

$\begin{array}{l}\frac{3+5i}{-1+2i}=\frac{3+5i}{-1+2i}\cdot \frac{-1-2i}{-1-2i}\\ =\frac{\left(3+5i\right)\left(-1-2i\right)}{\left(-1+2i\right)\left(-1-2i\right)}\\ =\frac{\left(3\right)\left(-1\right)+\left(3\right)\left(-2i\right)+\left(5i\right)\left(-1\right)+\left(5i\right)\left(-2i\right)}{{\left(-1\right)}^2-{\left(2i\right)}^2}\\ =\frac{-3-6i-5i-10i^2}{1-4i^2}\\ =\frac{-3-11i-10\left(-1\right)}{1-4\left(-1\right)}\\ =\frac{-3-11i+10}{1+4}\\ =\frac{7-11i}{5}\\ =\frac{7}{5}-\frac{11}{5}i\end{array}$

The quotient is $\frac{7}{5}-\frac{11}{5}i$ .