We can start by factoring the numerator and the denominator.  Starting with the numerator, we need to find two numbers that multiply to get  and add to get .  It happens that 3 and 3 work, so:

For the denominator, we need two numbers that multiply to get , and add to get .  Here,  and  work, so:

If we needed to, we always could have used the quadratic formula to factor each of the above equations.  We can now put the factored equations back in to our original function:

Then we can cancel the  terms.

In the beginning, we can treat the polynomials in the numerator and denominator as separate functions. We can use the quadratic formula for each, if we can't come up with a clever guess at the solution:

Returning the solutions back into the original function gives:

We cancel the  terms and have:

To factor we need to find two numbers that add to get , and multiply to get .  Because the  has a negative sign, we can take the factors of , which are  and  and see which would subtract to get to .  Due to the  being positive, we know that the larger number in each factoring pair would have to also be positive.  We can see that  would not equal , but  does, so:

While the  term in the denominator might be off-putting, let's start with the numerator and see where things go. We can try and factor the numerator by finding two numbers that add up to , and multiply to get .  It just happens that  and  do that.  Now our equation looks like this:

We can now cancel the  from the denominator for:

And then solve:

Check if the problem factors using the following property,

where c will be the sum of a and b, and d will be the product of them. We can prove this by expanding out the right hand side:

Therefore:     and:  

In our problem, -4 is the sum of a and b, and 4 is the product of a and b. What numbers for a and b can we use to satisfy our equation? Well, the numbers will have to be factors of 4, so we can start by writing these down: 1*4   and 2*2 Trying these out we can clearly see that the proper numbers are 2, and 2, since they are the only ones that add up to 4! Now that you have a and b, plug these back into the original relationship to find:

Expand this out for yourself to see that the property holds true!

Check if the problem factors using the following property,

where c will be the sum of a and b, and d will be the product of them. We can prove this by expanding out the right hand side:

Therefore:     and:  

In our problem, 10 is the sum of a and b, and 24 is the product of a and b. What numbers for a and b can we use to satisfy our equation? Well, the numbers will have to be factors of 24, so we can start by writing these down: 1*24  2*12   3*8    and 4*6. Trying a few of these out we can clearly see that the proper numbers are 4, and 6, since they are the only ones that add up to 10! Now that you have a and b, plug these back into the original relationship to find:

Expand this out for yourself to see that the property holds true!

Here we would need two numbers that multiply to get to , but would cancel each other when added.  That means that the number is going to be the same, but one will be positive and one will be negative.  Looking through the factors of , , we see that  multiplied by , gives us , which is what we want.  We then make one  positive, and the other negative:

Here we start to factor by first grouping the terms together:

From there we can see that we can factor an  out of the first group of terms, and a  from the second group of terms:

Now, because both groups share a common , we can factor that out of both of them:

Just to make sure we're finished, we can check to see if any of the remaining groups can be factored.  None can, so we're done.

To factor, let's begin by grouping the first two terms and the last two terms:

We can then factor out a common  from the first grouping, and a  from the second grouping:

As we can see, there's a common  from each group, and we can factor that out as well:

From here, we can double check that none of the terms can be factored further.  They can't be, so we're done.

To begin, let's group the first two terms:

We can easily see that we can factor out a  from the first group, giving us:

What's tougher to see is that we can also factor out a  from the second grouping, giving us:

Now we have a  in both groups, which we can factor it out:

We're not quite done yet though.  We can still factor:

Giving us:

We then make things look slightly nicer by collecting all the like terms for a final answer of: