Answer
$-\displaystyle \frac{5}{26}-\frac{51}{26}i$
Work Step by Step
$\displaystyle \frac{(5-2i)+(5+3i)}{(1+i)-(2-4i)}\qquad$ ...remove parentheses using the distributive property.
$=\displaystyle \frac{5-2i+5+3i}{1+i-2+4i}\qquad$ ...add like terms.
$=\displaystyle \frac{10+i}{-1+5i}\qquad$ ...rationalize by multiplying both the numerator and denominator with $-1-5i$.
$=\displaystyle \frac{10+i}{-1+5i}\cdot\frac{-1-5i}{-1-5i}$
$=\displaystyle \frac{(10+i)(-1-5i)}{(-1+5i)(-1-5i)}\qquad$ ...use the FOIL method.
$=\displaystyle \frac{-10-50i-i-5i^{2}}{1+5i-5i-25i^{2}}\qquad$ ...simplify and add like terms ($i^{2}=-1$).
$=\displaystyle \frac{-5-51i}{26}\qquad$ ...write in standard form $a+bi$
$=-\displaystyle \frac{5}{26}-\frac{51}{26}i$
