Question 281528
Starting from easiest to most complicated:

Greatest common factoring:
Take out the greatest common factor:
{{{5x + 5}}}
Since 5 is in both:
{{{ 5(x + 1)}}}

Trinomial factoring (easy):
This example will have nothing near the x^2
{{{x^2 + 3x + 2}}}

You need to find two things that add up to 3 and multiply to be 2.
In this case, 1 and 2 are your answers because 1 + 2 = 3 and 1 * 2 = 2

so {{{x^2 + 3x + 2 = (x+2)(x+1)}}}


Factor by grouping:
This will not work all the time.
{{{x^3 + x^2 + x + 1}}}
Group the first two terms and the last two terms:
{{{(x^3 + x^2) + (x + 1)}}}
We can use greatest common factoring, x^2 is the same in the first two terms and 1 is the same in the next two.

{{{x^2(x+1) + 1(x+1)}}}
At this step, you can tell if grouping will work, you will see that we have two (x+1), from this step, you can factor out an (x+1) from each term:

{{{(x^2 +1)(x+1)}}} many people see this step as taking only one of the (x+1) terms and taking the x^2, and the +1 and put it together.


Trinomial factoring (hard):
This example will have something near the x^2
{{{9x^2 + 6x + 1}}}

The method to do this one is to multiply the first and last number together:
{{{ 9 * 1 = 9}}}

Now you want to do something similar to the first one, two things that multiply to be 9 and add up to be 6. The answer is 3 and 3, so we divide the middle up like this:

{{{9x^2 + 3x + 3x + 1}}}

If all went right, you can factor by grouping:

{{{ (9x^2 + 3x) + (3x + 1) = 3x(3x +  1) + 1(3x + 1) = (3x + 1)(3x + 1) = (3x + 1)^2}}}

Those should be all the factoring methods you will need to know, the other one is much tougher but is very tricky and is taught at higher levels.