I have two examples of dividing complex numbers, but both do the sign differently.
The first is:
$$\frac{a+bi}{c-di} \cdot \frac{c+di}{c-di}$$
the other is:
$$\frac{2-3i}{5+2i}\cdot\frac{5-2i}{5-2i}$$
In the first the signs match the signs next to them, then in the second they are both $-$. How do I know what signs to use for this?
Multiply by the conjugate of the denominator.
$$ \frac{2-3i}{5+2i}\cdot\frac{5-2i}{5-2i}=\frac{10-19i-6}{25+4}=\frac{4-19i}{29} $$