Question 114553
For f(x) to be positive, the denominator has to stay greater than zero. 
{{{x^2-2x-8>0}}}
You can factor the denominator, 
{{{x^2-2x-8=(x-4)(x+2)}}}
The zeros are then x=-2 and x=4.
Let's break up the problem into three parts
1. {{{x<-2}}}
2. {{{x>4}}}
3. {{{-2<x<4}}}
That then covers the entire number line. 
Let's look at each region, 
1. {{{x<-2}}}
{{{(x-4)<0}}}
{{{(x+2)<0}}}
Negative times a negative equals a positive. 
f(x) is positive when {{{x<-2}}}.
2.{{{x>4}}}
{{{(x-4)>0}}}
{{{(x+2)>0}}}
Positive times a positive equals a positive. 
f(x) is positive when {{{x>4}}}.
3.{{{-2<x<4}}}
{{{(x-4)<0}}}
{{{(x+2)>0}}}
Negative times a positive equals a negative. 
f(x) is negative when {{{-2<x<4}}}.
The interval where f(x) is positive is ({{{x<-2}}}) and ({{{x>4}}}).