Answer
$\frac{20}{13}-\frac{30}{13}i$.
Work Step by Step
The given function is
$\Rightarrow f(x)=\frac{x^2+19}{2-x}$
Replace $x$ with $3i$.
$\Rightarrow f(x)=\frac{(3i)^2+19}{2-(3i)}$
Simplify.
$\Rightarrow f(x)=\frac{9i^2+19}{2-3i}$
Use $i^2=-1$.
$\Rightarrow f(x)=\frac{-9+19}{2-3i}$
Simplify.
$\Rightarrow f(x)=\frac{10}{2-3i}$
The conjugate of the denominator is $2+3i$.
Multiply the numerator and the denominator by $2+3i$.
$\Rightarrow f(x)=\frac{10}{2-3i}\cdot \frac{2+3i}{2+3i}$
Use the special formula $(A+B)^2=A^2+2AB+B^2$
$\Rightarrow f(x)=\frac{20+30i}{(2)^2-(3i)^2}$
Use $i^2=-1$.
$\Rightarrow f(x)=\frac{20+30i}{4+9}$
Simplify.
$\Rightarrow f(x)=\frac{20+30i}{13}$
Rewrite as $a+ib$.
$=\frac{20}{13}-\frac{30}{13}i$.
