📖 Factoring Is Reversing Multiplication
You've already learned how to multiply two binomials using the box method. Factoring is the same process — just run backwards.
When you see x² + 5x + 6, you're looking at the result of a multiplication. Your job is to figure out what was multiplied to create it.
Both processes use the same box. Multiplication fills the box going forward. Factoring fills the box going backwards.
📦 The Box Method — Four Steps
We use the same 2×2 box you've seen for multiplying binomials. The four cells each have a name and a job. Work through them in order.
Example: Factor x² + 5x + 6
Step 1
Variable Box
Put x² in the upper-left.
x² comes from x · x, so the outside factors are both x.
Step 2
Constant Box
Put 6 in the lower-right — the Constant Box.
Find pairs that multiply to 6: try 1×6, 2×3.
Step 3
Middle Term
Fill the middle boxes. 2x + 3x = 5x ✓
Matches the middle term!
Answer
Read the Factors
Read factors from the outside:
(x + 2)(x + 3)
The box has four named cells: the Variable Box (upper-left, always x²), the Constant Box (lower-right, the last number), and the two Middle Boxes (upper-right and lower-left, which must add to the middle term).
The order matters: Fill the Variable Box first → then the Constant Box → then use the answer choices to figure out the Middle Boxes last. The middle term is the trickiest part — that's where answer choices save you time.
± Sign Patterns — The Key to Factoring
The signs of the constant and middle term tell you exactly what signs to use in the factors. Learn these three patterns and you'll eliminate wrong answers instantly.
| Constant |
Middle Term |
Sign Pattern |
Example |
| Positive |
Positive |
Both positive
(x + ?)(x + ?) |
x² + 5x + 6 = (x+2)(x+3) |
| Positive |
Negative |
Both negative
(x − ?)(x − ?) |
x² − 7x + 12 = (x−3)(x−4) |
| Negative |
Either |
One positive, one negative
(x + ?)(x − ?) |
x² + x − 12 = (x+4)(x−3) |
GED strategy: Check the sign of the constant first. If it's negative, immediately eliminate any answer choices that use the same sign in both factors.
🎯 GED Strategy — Start with the Answer Choices
Many GED factoring questions are multiple choice — and when they are, the answer choices are your best tool. You don't have to factor from scratch. You can multiply the choices back out and check.
We'll practice that way first, with answer choices provided. Once you're comfortable, you'll be able to factor without them too.
Multiplying back is not guessing — it's verification. Factoring and multiplying are reverse processes. Testing a choice by multiplying it back out is exactly the right mathematical move.
The middle term is usually the only difference between two close answer choices. For example, (x + 2)(x + 3) and (x + 1)(x + 6) both give a positive constant of 6 — but their middle terms are 5x vs 7x. That's the question.
🔢 When a Monomial Was Multiplied First
Sometimes an expression was created by multiplying a number or variable times a larger expression. When you see all terms sharing a common factor, that's the signal.
Example 1
3x + 6 = 3(x + 2)
3 was multiplied times (x + 2). Both terms divide evenly by 3.
Example 2
6x² + 9x = 3x(2x + 3)
3x was multiplied times (2x + 3). Check: both 6x² and 9x divide evenly by 3x.
Always verify by multiplying back: 3x(2x + 3) = 6x² + 9x ✓
Factor completely means keep going until nothing can be factored further. Some expressions need two steps: first pull out the monomial, then factor the remaining trinomial.
🔢 Worked Examples
Every example below is presented with answer choices so you can practice the verification strategy. This is how many GED factoring questions appear — and it's a great way to build confidence with the box method.
Example 1 — Basic Positive Trinomial
Which expression correctly factors
x² + 5x + 6?
A) (x + 1)(x + 6)
B) (x + 2)(x + 3) ✓
C) (x − 2)(x − 3)
D) (x + 5)(x + 1)
Constant is +6, middle is +5x → both signs positive. Eliminate C immediately.
Try B: build the box for (x + 2)(x + 3):
Middle boxes: 2x + 3x = 5x ✓ — matches!
Answer: B — (x + 2)(x + 3)
Example 2 — Negative Middle Term
Which expression correctly factors
x² − 7x + 12?
A) (x + 3)(x + 4)
B) (x − 2)(x − 6)
C) (x − 3)(x − 4) ✓
D) (x − 1)(x − 12)
Constant is +12, middle is −7x → both signs negative. Eliminate A immediately.
Constant Box = 12. Test C — pairs (−3)(−4): build the box for (x − 3)(x − 4):
Middle boxes: −3x + (−4x) = −7x ✓ — matches!
Answer: C — (x − 3)(x − 4)
Example 3 — Negative Constant
Which expression correctly factors
x² + x − 12?
A) (x − 4)(x − 3)
B) (x + 4)(x + 3)
C) (x − 4)(x + 3)
D) (x + 4)(x − 3) ✓
Constant Box is −12 → one positive, one negative. Eliminate A and B — same signs in both.
Test D: build the box for (x + 4)(x − 3):
Middle boxes: +4x + (−3x) = +x ✓ — matches!
Answer: D — (x + 4)(x − 3)
Example 4 — Leading Coefficient (GED Level)
Which expression correctly factors
2x² + 7x + 3?
A) (2x + 1)(x + 3) ✓
B) (2x + 3)(x + 1)
C) (x + 3)(x + 1)
D) (2x + 7)(x + 1)
Variable Box = 2x² → outside factors are 2x and x. Eliminate C (gives only x², not 2x²).
Test A: build the box for (2x + 1)(x + 3):
Middle boxes: x + 6x = 7x ✓ — matches!
Compare to B: (2x+3)(x+1) → middle boxes = 2x + 3x = 5x ✗
Answer: A — (2x + 1)(x + 3)
Example 5 — Monomial Factoring
Which expression correctly factors
6x² + 9x?
A) 2x(3x + 4)
B) 3x(2x + 3) ✓
C) 6x(x + 3)
D) 3(2x² + 3x)
Both terms share 3x — it was multiplied times something. Use a 1-row box:
3x × 2x = 6x² | 3x × 3 = 9x ✓
Check others: A: 2x(3x+4) = 6x² + 8x ✗ | C: 6x(x+3) = 6x² + 18x ✗
Answer: B — 3x(2x + 3)
Example 6 — Factoring Out a Common Factor
Which expression correctly factors
4x² + 8x?
A) 2x(2x + 4)
B) 4x(x + 2) ✓
C) 4(x² + 2)
D) 2(2x² + 4x)
Both terms share a factor. What appears in both 4x² and 8x? → 4x
Use a 1-row box — 4x goes on the outside left, the two terms fill the boxes:
4x × x = 4x² | 4x × 2 = 8x ✓
Check others: A: 2x(2x+4) = 4x²+8x ✓ but (2x+4) still has a factor of 2 — not fully factored.
D: 2(2x²+4x) = 4x²+8x ✓ but 2x²+4x still has a common factor 2x — not done.
Answer: B — 4x(x + 2)
🚫 Common Mistakes
Mistake 1
Wrong signs despite correct numbers
x² − 5x + 6 ≠ (x + 2)(x + 3)
The signs should be (x − 2)(x − 3)
Mistake 2
Numbers multiply correctly but don't add correctly
For x² + 5x + 6: (1)(6)=6 ✓ but 1+6=7 ✗
Must check BOTH product AND sum
Mistake 3
Stopping before fully factored
(2x + 4)(x + 3) looks factored but
(2x + 4) = 2(x + 2) — not done yet
Mistake 4
Not using answer choices strategically
On the GED, multiply the choices back out.
This is smart test-taking, not guessing.
✏️ Practice Questions