We notice that this is the difference of two squared numbers:  and .

Hence, we can follow the rule that the difference of two perfect squares  is equal to.

To see this a little better, we can FOIL out the answer:

the s cancel out and we're left with the original equation:

Remember that whenever there's a problem involving factoring, you can always expand your answer again and see if you end up with the original expression given.

The first thing to notice is that the polynomial  has a common factor of  so we can factor it out automatically.

From here, we have a reducible quadratic factor in the parentheses. We know this because we consider the middle term: Half of the middle term squared is equal to the last term. Let's see this together: half of  the middle term, , is .  is  and equal to the last term.

 That means that we can factor the polynomial thusly:

To check to see if our answer is correct, we can expand it again to see if we end up with the original polynomial.

Expanding the two linear factors using FOIL

Distributing out the 9 in front, we have the original polynomial.

This one's tricky. We must pull out the greatest common factor from the polynomial first to see what we end up with. It looks like each of the terms has a factor of , , and . That means we can pull out  from each factor and put it in front of the parentheses.

Now, we can see that there's a quadratic factor that can be simplified. The polynomial in the parentheses can be easily factored because it is of a special class of quadratics: half of the middle number squared is equal to the last number**.

Which is our answer.

Remember, to check any factoring problem, one can expand the terms using the distributive property to see if the end result is the original polynonmial.

** In case there's some confusion about what I meant about the quadratic factor, consider this:

  is our quadratic. half of the middle number  equals . And  which is equal to the last term.

This whole process is similar to "completing the square". 

Factor out the largest common quantity:

Which two numbers can add/subtract to the middle term, but multiply to equal the last term?

The product of negative 8 and negative 5 is positive 40. Their difference is also negative 13.

Factor out the largest quantity common to all terms:

Factor the simplified quadratic:

You need to use the sum of two cubes equation

To factor a polynomial that has a coefficient in front of the  term, follow the steps below;

1) Once the equation is in standard form () , multiply the  term by the  term

2) Find two factors of this term that give you the  term

3) Re-write the polynomial with the original  term expanded into the two factors

4) Factor by grouping

5) Distribute to check that the factorization is correct

You need to factor by grouping but the important step is to remember the difference of squares.

To factor this, use trial and error to see what works. Since we have a  as the leading coefficient, it's helpful to remember that there's only one way to get  . Same with  as our third term--there's only one way to get  . Use these facts as you try to factor. Remember that signs matter. Therefore, your answer is: .

The best method for factoring this polynomial is by grouping: