Answer
$-5+i$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To divide the given expression, $
\dfrac{-38-8i}{7+3i}
,$ 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{-38-8i}{7+3i}\cdot\dfrac{7-3i}{7-3i}
\\\\=
\dfrac{(-38-8i)(7-3i)}{(7+3i)(7-3i)}
.\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{(-38-8i)(7-3i)}{(7)^2-(3i)^2}
\\\\=
\dfrac{(-38-8i)(7-3i)}{49-9i^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{-38(7)-38(-3i)-8i(7)-8i(-3i)}{49-9i^2}
\\\\=
\dfrac{-266+114i-56i+24i^2}{49-9i^2}
\\\\=
\dfrac{-266+(114-56)i+24i^2}{49-9i^2}
\\\\=
\dfrac{-266+58i+24i^2}{49-9i^2}
.\end{array}
Since $i^2=-1,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{-266+58i+24(-1)}{49-9(-1)}
\\\\=
\dfrac{-266+58i-24}{49+9}
\\\\=
\dfrac{-290+58i}{58}
\\\\=
\dfrac{58(-5+i)}{58}
\\\\=
\dfrac{\cancel{58}(-5+i)}{\cancel{58}}
\\\\=
-5+i
.\end{array}
