Answer
$\displaystyle \frac{ac+bd}{c^{2}+d^{2}}+\frac{bc-ad}{c^{2}+d^{2}}i$
Work Step by Step
$\displaystyle \frac{a+bi}{c+di}\qquad$ ...rationalize by multiplying both the numerator and denominator with $c-di$.
$=\displaystyle \frac{(a+bi)(c-di)}{(c+di)(c-di)}\qquad$ ...use the FOIL method
$=\displaystyle \frac{ac-adi+bci-bdi^{2}}{c^{2}-cdi+cdi-d^{2}i^{2}}\qquad$ ...simplify ($i^{2}=-1$).
$=\displaystyle \frac{ac-adi+bci+bd}{c^{2}+d^{2}}\qquad$ ...group the real and imaginary parts in the numerator.
$=\displaystyle \frac{ac+bd+(bc-ad)i}{c^{2}+d^{2}}\qquad$ ...write in standard form $a+bi$
$=\displaystyle \frac{ac+bd}{c^{2}+d^{2}}+\frac{bc-ad}{c^{2}+d^{2}}i$
