Answer
$-\displaystyle \frac{59}{106}-\frac{21}{106}i$
Work Step by Step
$\displaystyle \frac{-1-6i}{5+9i}\qquad$ ...multiply both numerator and denominator by $5-9i$
$=\displaystyle \frac{(-1-6i)(5-9i)}{(5+9i)(5-9i)}\qquad$ ...use the FOIL method.
$=\displaystyle \frac{-5+9i-30i+54i^{2}}{25-45i+45i-81i^{2}}\qquad$ ...simplify. ($i^{2}=-1$)
$=\displaystyle \frac{-5+9i-30i-54}{25-45i+45i+81}\qquad$ ...add like terms.
$=\displaystyle \frac{-59-21i}{106}\qquad$ ...write in standard form
$=-\displaystyle \frac{59}{106}-\frac{21}{106}i$
