The first thing to do when dividing polynomials is to look for and remove any common monomial factors. In terms of rational expressions, "looking for common factors" means that we're looking for those that go into every term in the numerator and into every term in the denominator. These are the factors we can pull out and cancel. At least it's way easier than canceling your LA Fitness membership. Those people are relentless.
The difference between what we did with common factors earlier and what we're doing now is that now we may cancel a common factor and still have a rational expression instead of a polynomial. That's fine. We can live with that, right? We've had more than our fill of polynomials lately anyway.
Sample Problem
What's the simplified version of the rational expression 
?
After some hunting, we can see that this guy has a common factor of x2 in the numerator and denominator. There are x's all over the place, but this is the most we can grab out of every single term. We pull this factor out of both the numerator and denominator.
And then we can cancel x2 from the top and bottom.
That's as simple as it gets this time. We've still got a rational expression on our hands, but at least it's a bit less intense-looking.