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2. The Greenebox Factoring Method
In Algebra, factoring has always been a very complicated procedure to learn as well as to teach. The main problem with the current way we teach factoring is there are too many methods to learn and each is used for a different type of polynomial. As if that weren’t complicated enough, there are at least seven formulas that need to be memorized as well.
The Greenebox Factoring Method can be used to factor polynomials with four terms, three terms and even two terms (the difference of two squares). The major advantage of this method is it gives the student ONE METHOD that can be used to factor almost everything. The only thing left to memorize are the cube formulas. Then after they feel confident with their factoring skills, the teacher can show them other shortcuts and formulas.
ã Copyright 1999 Lynda Greene
all rights reserved

3. Factoring Trinomials:
1. Split the middle term
2. Factor using the box

4. This is called “splitting the middle term”.
The Greenebox factoring method uses a box with four spaces in it. That means it works on polynomials that have 4 terms.
In other words:
A polynomial with 4-terms can be put into the box and factored immediately.
A polynomial with 3-terms can be put into the box after we change the three terms back into the original four.
This is called “splitting the middle term”.

5. Why do some polynomials have 3 terms and others have 4?
When the FOIL multiplication method
is used to multiply 2 binomials,
for example: (x + 3)(4x - 2) or (a + b)(c - d)
Each answer will have four terms (First, Outer, Inner, Last)
If the Outer and Inner terms can be combined,
the result will be a Trinomial (3 terms).
(x + 3)(4x - 2)= x2 - 2x + 12x - 6 = x2 + 10x - 6
If they cannot be combined, it keeps all 4 terms.
(a + b)(c - d) = ac - ad + bc - bd
An earlier lesson covered the process for factoring 4-terms, this tutorial covers factoring 3-term polynomials.

6. II. Factoring a Trinomial of the form: ax2 + bx + c
3 terms: 2x2 + 9x + 10
First
(a)
Middle
(b)
Last
(c)
First, we will split the middle term into the original outer and
inner terms. This gives us a four term polynomial which we
can factor using the Greenebox method.
This is the other application for the 2-dimensional factoring method. The first part of this procedure is borrowed from a factoring method called the ac-method. The ac-method, splits the middle term into the original outer and inner terms then uses grouping. Since the box is an alternative to grouping, we use the ac-method to split the middle term, then use the box instead of grouping to factor.
There are two examples here. One with a positive last term and one with a negative last term.

7. Step 1: Multiply first*last Step 4: Which pair of factors gives us
2x2 + 9x + 10
first
middle
last
Splitting the middle term
Step 1: Multiply first*last
2 x 10 = 20
Step 3: The sign of the last term tells us whether to add or subtract the factors of 20.
Add 20
Step 2: Find all the
FACTORS of 20
1  20
2  10
4  5
= 21
= 12
= 9
Step 4: Which pair of factors gives us
the middle term?
Since the middle term is 9x, the original 4 and 5 each had an“x”.
(They were like terms and were added together)
Step 5: Choose the correct signs.
FACT: We add numbers when they have the same signs. (step 3).
i.e. The factors are both positive, + 4x and + 5x OR both negative, -4x and -5x
This is the step-by-step method of splitting the middle term. Again, this is a lot shorter when you actually use it.
1. Multiply the first times the last term
2. Find all the factors of the product
3. Look at the sign of the last term to see if you should add or subtract these pairs of factors
4. Pick the factors that give you the correct middle term
5. Choose the correct signs: If you added the factors, both factors have the sign of the middle term
6. If you subtracted the factors, one will be positive and the other negative
(The bigger number will have the same sign as the middle term)
IMPORTANT NOTE: SINCE THE TWO FACTORS WE HAVE CHOSEN ADD UP TO THE MIDDLE TERM WHICH HAS AN ‘x’ IN IT, THEY EACH ALSO HAVE AN ‘x’.
+ 4x and + 5x = + 9x
This pair gives us the correct middle term.

8. 1. Split the middle term into 2 terms
Note: We did step 1 on the previous page
2x2 + 9x + 10
1. Split the middle term into 2 terms F O I L
2x2 + 4x + 5x + 10
2. Place each term in the correct location in the box x
+ 2 F O I L 2x
2x2
+ 4x
3. Factor
out GCF
for each
row &
column
4. Answer:
(x + 2)(2x + 5)
Now that the polynomial has been returned to it’s original four term status, we can factor it using the box
1. Draw the box
2. Place the 4 terms in the correct locations, with their signs
* Remember that the two middle terms can go in either position, but the first and last terms must go on the top left and bottom right, respectively*
3. Factor out the GCF for each row and column, taking the sign of the middle terms.
4. Write these in the form of a product +5
+ 5x
+ 10

9. Step 3: Pick the pair that subtract to equal -10x
Subtraction
example
first
last
Step 1: Multiply first*last
3x2 - 10x - 8
SUBTRACT
3 x 8 = 24 24
Step 2. Find all the factors of 24
1 * 24
2 * 12
3 * 8
4 * 6
= 23
= 10
= 5
= 2
Step 3: Pick the pair that subtract to equal -10x
(the middle term)
This pair works
Step 4: Pick the correct signs:
(Subtract means: different signs)
+12x - 2x = +10x
- 12x + 2x = -10x
Here is an example of the same process but the last term is negative
1. This page shows how to split the middle term when subtracting
correct terms: -12x and +2x

10. 3. Factor out GCF for each row & column
3x2 - 10x - 8
1.Split the middle
term into 2 terms F O I L
3x2 + 2x - 12x - 8
2. Place each term in the correct location in the box 3x +2 F O I L x
3x2
+2x
3. Factor out GCF for each row & column
4. Answer:
(3x + 2)(x - 4)
2. Draw the box
3. Place the four terms in the correct location inside the box
4. Factor out the GCF, taking the signs from the middle terms
5. Write the answer in the form of a product -4
- 8
- 12x

11. x2 +3x - 10 - Note:first= 1 Now factor it -2x and +5x = +3x
II. Factoring a Trinomial of the form: x2 + bx - c
Note:first= 1
x2 +3x - 10 -
Step 3:
Subtract
Step 1: Multiply
the first*last
1 * 10 = 10
1 * 10
2 * 5
= 9
= 3
Step 2: Find the factors of 10
Step 4: Choose the pair of factors that equal + 3x (the middle term)
Step 5: Choose the correct signs:
-2x and +5x = +3x
or -5x and +2x = -3x
Usually, after letting the students work about three of these problems, I point out to them that if they use only the last term, they get the same product
When subtracting, the signs will be different
(One “+” , the other “-”)
Now factor it

12. Factor out GCF for each row & columnx2
+ 3x
- 10
Split the middle
term into 2 terms F O I L x2
+ 5x
- 2x
- 10
Place each term in the correct location in the box x +5 F O I L x 2 x
+ 5x
Factor out GCF for each row & column
Answer:
(x + 5)(x - 2)
I then point out to them that the numbers that they put into the box (in the outer and inner slots) are the same ones they eventually factored out. So, if they take those two factors and place them directly into the answer, they can bypass the use of the box. (ONLY IF THE COEFFICIENT OF THE FIRST TERM IS A 1)
- 2
- 2x
- 10

13. SPECIAL CASE!!!
There are some polynomials that have a factor that
must be taken out before using the box.
Take a polynomial such as:
4x2 + 10x - 6
This polynomial has a common factor of “2” in all three terms. If this is not taken out before using the box, the “2” will be taken out twice, doubling the answer. (Example on the next two slides)
This is a useful feature of the Greenebox Method. It forces the student to take out the GCF BEFORE trying to factor the terms by catching extra factors which might otherwise go unnoticed. If an extra factor is not taken out at some point, the answer will be wrong.

14. Step 4: Choose the pair of factors that equal + 10x
The way this problem should NOT be worked
This problem has a GCF of “2”. We’ll pretend we didn’t see it and work the problem normally,
4x2 + 10x - 6 -
Step 3:
Subtract
4 * 6 = 24
Step 1: Multiply
the first*last
1 * 24
2 * 12
3 * 8
4 * 6
= 23
= 10
= 5
= 2
Step 2: Find the factors of 24
Step 4: Choose the pair of factors that equal + 10x
(the middle term)
Step 5: Choose the correct signs:
-2x and +12x = +10x
or 2x and -12x = -10x

15. 4x2 - 2x - 6 2x + 6 4x + 12x - 2 (2x + 6)(4x - 2)= 8x2 + 20x - 12X
The way this problem should NOT be worked
4x2 + 12x - 2x - 6
Put the four factors into the box and factor it 2x
+ 6
If we check
the answer using FOIL, we
DO NOT get
the original problem!!!
4x2 + 10x - 6
The box
“SEES”
the “2”
in both
dimensions
and pulls it
out twice!!! 4x
4x2
+ 12x
- 2
- 2x
- 6
When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term.
(2x + 6)(4x - 2)= 8x2 + 20x - 12X
WRONG ANSWER!!!!!! THIS MEANS THERE WAS A FACTOR WE MISSED!!!

16. Step 4: Choose the pair of factors that equal + 5x
The way this problem SHOULD be worked
This problem has a GCF of “2”. We’ll factor it out then work the problem normally using only the trinomial (in parentheses). -
Step 3:
Subtract
4x2 + 10x - 6 = 2 (2x2 + 5x - 3)
Step 1: Multiply
the first*last
2 * 3 = 6
1 * 6
2 * 3
Step 2: Find the factors of 6
= 5
= 1
Step 4: Choose the pair of factors that equal + 5x
(the middle term)
Step 5: Choose the correct signs:
-x and +6x = +5x
or x and -6x = -5x

17. 2x2 + 6x - x - 3 x + 3 2x - 1 2(x + 3)(2x - 1)= 4x2 + 10x - 6 
The way this problem should have been worked
2 (2x2 + 6x - x - 3)
Example: x
+ 3
Now, place
the four terms
into the box
and factor
normally
2x2 2x
+ 6x
Don’t forget to
put that extra
“2” in the answer!!!
- 1
- x
- 3
When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term.
2(x + 3)(2x - 1)= 4x2 + 10x - 6 

18. 4x2 - 5x - 6 3y2 - 16y + 5 2x2 + 9x - 18 Answers: 1. (4x + 3)(x - 2)
Some practice problems
Answers:
1. (4x + 3)(x - 2)
2. (3y - 1)(y - 5)
3.(2x - 3)(x + 6)
4. (x + 1)(x + 2)
5. (x - 6)(x + 2)
4x2 - 5x - 6 1. 2. 3.
3y2 - 16y + 5
2x2 + 9x - 18
4. x2 + 3x + 2
5. x2 - 4x - 12
Make sure to check for overall GCF’s before using the box.

19. Questions? send e-mail to: lgreene1@satx.rr.com
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