Answer
$\dfrac{7+17i}{13}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To divide the given expression, $
\dfrac{-1+5i}{3+2i}
,$ multiply both the numerator and the denominator by the complex conjugate of the denominator. Use special products to multiply the resulting expression and use $i^2=-1.$
$\bf{\text{Solution Details:}}$
Multiplying both the numerator and the denominator by the complex conjugate of the denominator, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{-1+5i}{3+2i}\cdot\dfrac{3-2i}{3-2i}
\\\\=
\dfrac{(-1+5i)(3-2i)}{(3+2i)(3-2i)}
.\end{array}
Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\dfrac{(-1+5i)(3-2i)}{(3)^2-(2i)^2}
\\\\=
\dfrac{(-1+5i)(3-2i)}{9-4i^2}
.\end{array}
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the expression above is equivalent to\begin{array}{l}\require{cancel}
\dfrac{-1(3)-1(-2i)+5i(3)+5i(-2i)}{9-4i^2}
\\\\=
\dfrac{-3+2i+15i-10i^2}{9-4i^2}
\\\\=
\dfrac{-3+(2+15)i-10i^2}{9-4i^2}
\\\\=
\dfrac{-3+17i-10i^2}{9-4i^2}
.\end{array}
Since $i^2=-1,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{-3+17i-10(-1)}{9-4(-1)}
\\\\=
\dfrac{-3+17i+10}{9+4}
\\\\=
\dfrac{7+17i}{13}
.\end{array}