These are the correct factors for each part of the fractions. I know factoring
can be tough, so if you're getting at least half of these right, great work. If
you got stuck on one of these, try to look back at your work and see where you
went wrong to get these factors. If you can't find an error in your work then
stay with me for this solution. For our first numerator we can pull out a 2x
from each term. So we'll have 2x times x squared minus 6x plus 9. Then we notice
that this is a special factoring pattern. It's a perfect square trinomial. So we
have 2x times x minus 3, times x minus 3. This gives us our first numerator. For
this denominator, we want to find the factors of negative 12, that sum to
negative 4. This allows us to rewrite our middle term. And then we'll use
factoring by grouping to get 3x plus 2 times x minus 2. This is our factored
form for our first denominator. For this numerator we want to find the factors
of 12 that sum to negative 8. These two factors are negative 6, 6 and negative
2. We'll use factoring by grouping to get 3x minus 2 time x minus 2 for this
numerator. And finally for this last denominator we pull out a 6x from both
terms leaving us with 6x times x squared minus 9. Then we can factor this
difference of squares using x plus 3 times x minus 3. This difference of squares
pattern appears in all sorts of math. So it's great that we can recognize it
quickly.