Answer
$\frac{11}{13}+\frac{10}{13}i$
Work Step by Step
$\displaystyle \frac{1+4i}{3+2i}$
We multiply the numerator and denominator by the conjugate of the denominator:
$=\frac{1+4i}{3+2i}\cdot\frac{3-2i}{3-2i}$
We use the fact that $i^2=-1$ (because $i=\sqrt{-1}$)
$=\frac{3-2i+12i-8*-1}{3^{2}+2^{2}}$
$=\frac{11+10i}{13}$
$=\frac{11}{13}+\frac{10}{13}i$
