Answer
$-3-2i$
Work Step by Step
Multiplying the numerator and the denominator by $2i,$ the given expression, $
\dfrac{4-6i}{2i}
,$ is equivalent to
\begin{align*}
&
\dfrac{4-6i}{2i}\cdot\dfrac{2i}{2i}
\\\\&=
\dfrac{4(2i)-6i(2i)}{(2i)^2}
\\\\&=
\dfrac{8i-12i^2}{4i^2}
\\\\&=
\dfrac{8i-12(-1)}{4(-1)}
&\text{ (use $i^2=-1$)}
\\\\&=
\dfrac{8i+12}{-4}
\\\\&=
\dfrac{8i}{-4}+\dfrac{12}{-4}
\\\\&=
-2i-3
\\\\&=
-3-2i
.\end{align*}
Hence, the given expression simplifies to $
-3-2i
$.
