Step 1 - Factor Out the GCF

Start by finding the Greatest Common Factor (GCF) of all terms and pulling it out:

6x³ + 9x² = 3x²(2x + 3)

Step 2 - Difference of Squares

If you have a² - b², factor it as (a+b)(a-b):

x² - 25 = (x + 5)(x - 5)
4x² - 9 = (2x + 3)(2x - 3)

Step 3 - Factoring Trinomials (x² + bx + c)

Find two numbers that multiply to c and add to b:

x² + 5x + 6 → find two numbers that multiply to 6 and add to 5
→ 2 and 3 → (x + 2)(x + 3)

Step 4 - Factoring Trinomials (ax² + bx + c)

Use the AC method: multiply a × c, find factors that add to b, then split the middle term and factor by grouping:

2x² + 7x + 3
ac = 6 → factors: 1 × 6, and 1 + 6 = 7 ✓
= 2x² + x + 6x + 3
= x(2x + 1) + 3(2x + 1)
= (x + 3)(2x + 1)

Step 5 - Sum & Difference of Cubes

a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)

Factoring by Grouping

For four-term polynomials, group terms in pairs and factor each group:

x³ + 2x² + 3x + 6
= x²(x + 2) + 3(x + 2)
= (x² + 3)(x + 2)

Checking Your Answer

Multiply your factors back together to confirm they equal the original polynomial. This is the simplest way to check your work.