Answer
$\frac{6}{5} + \frac{8}{5}i$
Work Step by Step
We want to eliminate imaginary numbers in the denominators of a rational function.
To do this, multiply both the numerator and the denominator by the conjugate of the denominator.
The conjugate of the denominator in this exercise is $2 + i$.
$=\dfrac{4 + 2i}{2 - i} \cdot \dfrac{2 + i}{2 + i}$
Combine into one fraction:
$=\dfrac{(4 + 2i)(2 + i)}{(2 - i)(2 + i)}$
Use the FOIL method to distribute terms:
$=\dfrac{8 + 4i + 4i + 2i^2}{4 + 2i - 2i - i^2}$
Combine like terms:
$\dfrac{8 + 8i + 2i^2}{4 - i^2}$
Simplify $i^2$ terms by replacing them with $-1$:
$\dfrac{8 + 8i + 2(-1)}{4 - (-1)}$
Multiply to simplify:
$\dfrac{8 + 8i - 2}{4 + 1}$
Combine like terms:
$\dfrac{6 + 8i}{5}$
Rewrite in a more conventional form by separating the fraction into its components:
$\frac{6}{5} + \frac{8}{5}i$
