When you multiply two binomials together, such as (x + 1) and (x - 2), you use the FOIL method, multiplying the First, Outside, Inside and Last terms and adding the results. In this case, that will give you x^2 - 2x + x - 2. This simplifies to the trinomial x^2 - x - 2. Factoring a trinomial is the opposite of this process. You want to find the two binomials that you could have multiplied together using FOIL to arrive at that trinomial.
Before you try to determine two binomials that multiply together to give you the trinomial, first see if there is a common factor to all your terms that you can factor out to simplify your expression . For example, if the trinomial expression is 2x^2 - 2x - 4, factor the two out: 2(x^2 - x - 2). If the leading term is negative, factor a -1 out: -x^2 + x + 2 = -1(x^2 - x - 2). Now you are left with the simple trinomial (x^2 - x - 2). There is a simple procedure to factor trinomials of this form, ax^2 + bx + c, where "a" is equal to one.
If the trinomial is of the form ax^2 + bx + c and "a" is equal to one, then set up your two binomials and put "x" as the first term of each: (x___) (x___). To find the second terms, find two numbers that, if multiplied together, will give you "c," and if added will give you "b." In the case of x^2 - x - 2, b is -1 and c is -2. The only numbers that add to -1 and multiply to 2 are -2 and 1. If the trinomial is x^2 - 7x + 10, then you need to find two numbers that add to -7 and multiply to 10. The numbers that satisfy this are -2 and -5. The factors are (x - 2) and (x - 5).
If you've already factored out your greatest common factor and are left with a trinomial of the form ax^2 + bx + c and "a" is not equal to 1, then determining the factors requires some trial and error. Suppose you had the trinomial 10x^2 - 19x - 2. You know that the 10x^2 is either the product of 10x and x, or 2x and 5x. Looking at the -2, you know that it must be the product of -2 and 1, or 2 and -1. Guessing that the first terms of the binomials are 2x and 5x, you can see that (5x - 2)(2x + 1) does not work. Switching the second terms, (5x + 1)(2x - 2) doesn't work either. 5x and 2x are not correct, so you try x and 10x. (10x - 2)(x + 1) still doesn't work. Switch the second terms: (10x + 1)(x - 2). Multiplying through with FOIL, you get your original trinomial.
Factoring polynomial expressions takes practice. Unlike multiplication with FOIL, you may have to use trial and error and make a few false starts before you hit upon the right combination. Be systematic and write out all your possibilities before you check them one at a time. As you practice more, you'll find that you will intuitively recognize patterns and will pick the correct combination more frequently, making fewer errors as you make more trials. There are no cheats for factoring trinomials. But if you are thorough, methodical and practice frequently, you'll never need to cheat.