UnFOILing is a method for factoring a trinomial into two binomials. When you multiply two binomials together, you use the FOIL method, multiplying the **F**irst, then the **O**uter, then the **I**nner, and finally the **L**ast terms of the two binomials into a trinomial. But when you need to factor a trinomial, you unFOIL by determining the factor pairs for *a *and *c,* the correct signs to place inside the two binomials, and what combination of factor pairs of *a *and *c* results in *b.*

The key to unFOILing is being organized:

Be sure you have an expression in the form:

Write the terms in the order of decreasing powers.

Remember how to assign the correct signs in each binomial:

The signs are

**both positive,**if*c*is positive and*b*is positive.The signs are

**both negative,**if*c*is positive and*b*is negative.**One sign is positive and one negative,**if*c*is negative; which binomial is positive and which one is negative depends on whether*b*is positive or negative and how you arranged the factors.

*Example: *

Determine all the ways you can multiply two numbers to get

*a.*You can get these numbers from the prime factorization of

*a*. Sometimes, writing out the list of ways to multiply is a big help. In this example,*a*is 24, and the list of ways you can multiply two numbers to get 24 is:1 × 24, 2 × 12, 3 × 8, or 4 × 6.

Determine all the ways you can multiply two numbers to get

*c.*In this example,

*c*is 45, and you can multiply the following numbers to get 45:1 × 45, 3 × 15, or 5 × 9.

Ignore the sign at this point. You don't need to worry about signs until Step 3.

Look at the sign of

*c*and your lists from Steps 1 and 2 to see if you want a*sum*or*difference.***If**find a value from your Step 1 list and another from your Step 2 list such that the sum of their product and the product of the two remaining numbers in those steps results in*c*is positive,*b.***If**find a value from your Step 1 list and another from your Step 2 list such that the difference of their product and the product of two remaining numbers from those steps results in*c*is negative,*b.*For the trinomial

*c*is negative, so you want a*difference*of 34 between products.Choose a product from Step 1 and a product from Step 2 that result in the correct sum or difference determined in Step 3.

Because you determined in Step 3 that you want a

*difference*of 34 between products, use 4 × 6 from*a*and 5 × 9 from*c.*The product of 4 and 5 is 20. The product of 6 and 9 is 54. The difference of these products is 34.

Arrange your choices as binomials so the results are those you want.

(4

*x**x*Place the signs to give the desired results.

(4

*x*– 9)(6*x*+ 5)FOIL the two binomials to check your work.

If the binomials are correct, you'll end up with the original problem when you FOIL them.