Factorise the following:

a(a - 1) - b(b - 1)

a(a - 1) - b(b - 1) = a² - a - b² + b

= a² - b² - a + b

= a² - b² - (a - b).

Using identity,

a² - b² = (a - b)(a + b).

a² - b² - (a - b) = (a - b)(a + b) - (a - b)

= (a - b)(a + b - 1).

Hence, a(a - 1) - b(b - 1) = (a - b)(a + b - 1).