So, what is a quadratic trinomial is?  Well, a trinomial is a polynomial with 3 terms.  A quadratic trinomial has the form:

ax2 + bx + c, in which a, b, and c are constants.

So, how can we factor that?  I will show an example.  For now, we will only work with quadratic trinomials in which a = 1, because they are much easier.

x2 + 5x + 6

To do this, we actually factor the expression into two binomials, instead of a monomial and a polynomial.  Can you figure it out?  Nah...I will just tell you: it is (x + 2)(x + 3)

*Keep in mind that the (x+2) and (x+3) can be switched because of the commutative property.*

To look for a pattern, we will multiply the binomials out, except in a vertical form (because I feel like it).

x + 3
* x + 2
2x + 6
x2 + 3x
x2 + 5x + 6

So it checks out.  But how on earth did I know it would? (I'm a psychic - just kidding).

You can see that the two constants (2 and 3) I put in the binomials multiplied to give 6, or "c", and they added to the coefficient of x, or "b."  So the rule is:

If the x2 coefficient is 1,
Find two numbers (say h and r) that multiply to give c and add to give b.  Then make two binomials in the form (x + h) and (x + r).

Yes, it's a little messy, but I'll do an example.

x2 + 7x + 12
First list some factors of 12:
12, 1    6, 2    3, 4    -4, -3 (remember you can multiply with negatives too).

Do any of these pairs have a sum of 7?  Yes, 3 and 4.

So we write x2 + 7x + 12 = (x + 3)(x + 4)

Let's do another example, except this time a little harder.
x2 + 4x - 21

First list factors of -21 (we take the sign in front of it to determine whether it's pos. or neg.)
-21, 1    -7, 3     -3, 7    -1, 21

Do any have a sum of 4?  Yes, -3 and 7.

So we write (x - 3)(x + 7).

Some trinomials cannot be factored using integers, example is x2 + 4x + 17.

If you want a little practice, try these:
1. x2 + 4x - 5
2. x2 - 3x + 2
3. x2 - 6x - 7
4. x2 + 4x + 4