Factorize: a³ – b³ + 1 + 3ab.

As the expression is of the form of sum of three cubes, for factorizing, we will use the identity

x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)

The expression a³ – b³ + 1 + 3ab is similar to the left hand side of the above identity (with x = a, y = –b and z = 1).

⇒ a³ – b³ + 1 + 3ab = (a)³ + (–b)³ + (1)³ – 3(a)(–b)(1)

= [a + (–b) + 1][a² + (–b)² + 1²– (a)(–b) – (–b)(1) – (1)(a)]

= (a – b + 1)(a² + b² + 1 + ab + b – a)

= (a – b + 1)(a² + b² + ab + a + b + 1)

Hence, the factors of a³ – b³ + 1 + 3ab are (a – b + 1)and (a² + b² + ab + a + b + 1).