• Factor out the greatest common factor, leaving ax² +bx +c.
• Find the product of a and c.
• Factor this product so that the sum of the factors is b. That is, find factors d and e of ac such that d + e = b.
• Write (ax + d) (ax + e).
• Factor out and DROP any common factor from each binomial factor.

SOLUTION:

• Notice that there is no common factor.
• Find the product 4(5) = 20
• List factors of 20: (1) (20), (2) (10), (4) (5)
• Find the factors whose sum is –21: -1 and –20.
• Write (4x – 1) (4x – 20).
• Drop the common factor from each binomial factor: (4x –1) (x –5).

EXAMPLE 2: Factor 6x² – 5x – 6.

SOLUTION:

• Notice that there is no common factor.
• Find the product 6(-6) = -36
• A. List factors of 36: (1) (36), (2) (18), (3) (12), (4) (9), (6) (6)
• Find the factors whose sum is –5: 4 and –9.
• Write (6x + 4) (6x – 9).
• Drop the common factor from each binomial factor: (3x + 2) (2x – 3).

EXAMPLE 3: Factor 12x² + 17x – 7.

SOLUTION:

• Notice that there is no common factor.
• Find the product 12(-7) = -84
• A. List factors of 84: (1) (84), (4) (21), (2) (42),  (6) (14), (3) (28),  (12) (7)
• Find the factors whose sum is 17: -4 and 21.
• Write (12x – 4) (12x + 21).
• Drop the common factor from each binomial factor: (3x – 1) (4x + 7).

EXAMPLE 4: Factor 8b⁵ – 68b⁴ + 84b³.

SOLUTION:

• Take out the common factor: 4b³ (2b² – 17b + 21).
• Find the product: (2) (21) = 42
• A. List factors of 42: (1) (42), (2) (21), (3) (14), (6) (7)
• Find the factors whose sum is –17: -3 and –14.
• Write 4b³(2b – 3) (2b – 14).
• Drop the common factor from each binomial factor: 4b³ (2b – 3) (b – 7)