
We can factor this numerator as the difference of two squares. We can write it

as 6 minus x times 6 plus x. For this denominator we want to find the factors of

negative 48 that sum to positive 2. These factors are negative 6 and positive 8.

When I look at my factors it doesn't seem as if anything can simplify out to

equal 1 but if i look at these two factors i can see that these are opposite

polynomials. The constant term 6 is positive while this one's negative and the x

term is negative here while this x term is positive. This means I'll have a

factor of negative 1. We can see this if we were to just switch the order of the

terms for these two, we'll have negative 6 plus x, then I can factor a negative

1 from this factor. I'll have negative 1 times positive 6 minus x. I change the

signs of both of these terms since i factored a negative 1 out. These factors of

6 minus X simplified to 1 but i still have a negative 1 out in front, just like

I thought here. Whenever you see opposite polynomial factors you can simplify

them to be negative 1. We don't need to show all this work. So we're left with

this expression. Now, I want to acknowledge there are a lot of different answers

here. This negative sign could have been distributed to either the numerator or

the denominator. If we distribute the negative 1 to the numerator, we could have

gotten this expression. Or switch the order of these terms and you got this

expression. These are all equal. If we distribute this negative one instead of

the denominator, we can wind up with any of these expressions. And in fact, all

of these are correct. In general, it's easiest to list the negative sign out in

front of a fraction and then write your rational expressions.