Answer
$-\displaystyle \frac{2}{5}+\frac{1}{5}i$
Work Step by Step
$\displaystyle \frac{(10+4i)-(3-2i)}{(6-7i)(1-2i)}\qquad$ ...remove parentheses in the numerator using the distributive property.
$=\displaystyle \frac{10+4i-3+2i}{(6-7i)(1-2i)}\qquad$ ...add like terms in the numerator.
$=\displaystyle \frac{7+6i}{(6-7i)(1-2i)}\qquad$ ...use the FOIL method in the denominator.
$=\displaystyle \frac{7+6i}{6-12i-7i+14i^{2}}\qquad$ ...simplify and add like terms ($i^{2}=-1$).
$=\displaystyle \frac{7+6i}{-8-19i}\qquad$ ...rationalize by multiplying both the numerator and denominator with $-8+19i$.
$=\displaystyle \frac{(-7-6i)(8-19i)}{(8+19i)(8-19i)}\qquad$ ...use the FOIL method in the numerator and $(a-b)(a+b)=a^{2}-b^{2}$ in the denominator.
$=\displaystyle \frac{-56+133i-48i+114i^{2}}{8^{2}-(19i)^{2}}\qquad$ ...simplify and add like terms ($i^{2}=-1$).
$=\displaystyle \frac{-170+85i}{64+361}$
$=\displaystyle \frac{-170+85i}{425}\qquad$ ...divide entire expression with $85$
$=\displaystyle \frac{-2+i}{5}\qquad$ ...write in standard form $a+bi$
$=-\displaystyle \frac{2}{5}+\frac{1}{5}i$
