Complementary angles are two angles whose measures add to 90°. The ratio of the measures of two complementary angles is 4:11.
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To factor both numerator and denominator in this rational expression we are going to substitute 
with 
; so 
and 
. This way we can rewrite the expression as follows:
Now we have two much easier to factor expressions of the form
. For the numerator we need to find two numbers whose product is 
(30) and its sum 
(-11); those numbers are -5 and -6. 
and 
.
Similarly, for the denominator those numbers are -2 and -5.
and 
. Now we can factor both numerator and denominator:
Notice that we have
in both numerator and denominator, so we can cancel those out:
But remember than
, so lets replace that to get back to our original variable:
Last but not least, the denominator of rational expression can't be zero, so the only restriction in the variable is
Now we have two much easier to factor expressions of the form
Similarly, for the denominator those numbers are -2 and -5.
Notice that we have
But remember than
Last but not least, the denominator of rational expression can't be zero, so the only restriction in the variable is