Factorise the following:

a² - 3a - 54

To factorise a² - 3a - 54, we want to find two real numbers whose sum is -3 and product is -54. By trial, we see that -9 + 6 = -3 and -9 × 6 = -54.

∴ a² - 3a - 54 = a² - 9a + 6a - 54

= a(a - 9) + 6(a - 9)

= (a - 9)(a + 6).

Hence, a² - 3a - 54 = (a - 9)(a + 6).