We look for the comon factors of each term in the polynomial (that means, the things that each term has in common).

It turns out that each term in the polinomial has a factor (multiplicative "part") of . In this case  could be a number or a variable but it doesn't matter in this case. To factor, we need to synthisize what's common with all the factors and put it in front of parentheses. Factoring like this helps to simplify equations and expressions down the road. 

Noticing that  is the common factor, we can take it outside of the parentheses, writing within what's left over after we take out the .

The polynomial  fits a special class of polynomials because the last number, , is the square of half the middle number, . i.e.  the middle number is the last number squared.

to see if it's right, we can expand it all over again:

usinig FOIL to expand the expression,

Which was our original polynomial.

We start with the polynomial

and, putting the summands in ascending order of degree, we have

We notice that the comon factor is  because it "fits" in each of the terms.

Notice

so, we factor  out and we have

.

 fits a common model of a special class of polynomial because the last number, , is half the middle term, squared (i.e. take  the middle number and square it and you have the last term in the polinomial.) This is very similar to the process of "completing the square" of a quadratic.

The goal is to factor out the greatest common factor to leave the polynomial in a much cleaner state. We notice that there is a factor of  and  (it could be that  is a number or a variable but, in this case, it doesn't matter). We can pull out  from all of the terms and put it in front.

Factoring out the 2b and leaving what's left inside of the parentheses, we get:

Note that this can't be simplified or factored anymore because there are no more common factors within the parentheses.

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: