Answer
$\dfrac{-23+43i}{58}$
Work Step by Step
Multiplying by the conjugate of the denominator, the given expression, $
\dfrac{4+5i}{3-7i}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4+5i}{3-7i}\cdot\dfrac{3+7i}{3+7i}
\\\\=
\dfrac{4(3)+4(7i)+5i(3)+5i(7i)}{(3)^2-(7i)^2}
\\\\=
\dfrac{12+28i+15i+35i^2}{9-49i^2}
\\\\=
\dfrac{12+28i+15i+35(-1)}{9-49(-1)}
\\\\=
\dfrac{12+28i+15i-35}{9+49}
\\\\=
\dfrac{(12-35)+(28i+15i)}{9+49}
\\\\=
\dfrac{-23+43i}{58}
.\end{array}
