Answer
$\dfrac{3+6i}{10}$
Work Step by Step
Multiplying by the conjugate of the denominator, the given expression, $
\dfrac{3i}{4+2i}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{3i}{4+2i}\cdot\dfrac{4-2i}{4-2i}
\\\\=
\dfrac{3i(4)+3i(-2i)}{4^2-(2i)^2}
\\\\=
\dfrac{12i-6i^2}{16-4i^2}
\\\\=
\dfrac{12i-6(-1)}{16-4(-1)}
\\\\=
\dfrac{12i+6}{16+4}
\\\\=
\dfrac{6+12i}{20}
\\\\=
\dfrac{\cancel{2}(3+6i)}{\cancel{2}\cdot10}
\\\\=
\dfrac{3+6i}{10}
.\end{array}
