Factorize: 8a³ – b³ – 12a²b + 6ab².

To factorize 8a³ – b³ – 12a²b + 6ab², we make use of the identity: (x – y)³ = x³ – 3x²y + 3xy² – y³

This expression 8a³ – b³ – 12a²b + 6ab² has four terms and is similar to the right hand side of the above identity.

Observe 8a³=(2a)³, 12a²b=3(2a)²(b) and 6ab²=3(2a)(b)²

⇒ 8a³ – b³ – 12a²b + 6ab² = (2a)³ – 3(2a)²(b) + 3(2a)(b)² – b³

∴ 8a³ – b³ – 12a²b + 6ab² = (2a – b)³

Hence, the factorization of8a³ – b³ – 12a²b + 6ab² is (2a – b)³.