Using factor theorem, show that (x - 2) is a factor of x³ + x² - 4x - 4. Hence, factorise the polynomial completely.

By factor theorem, (x - a) is a factor of f(x), if f(a) = 0.

f(x) = x³ + x² - 4x - 4

(x - 2) is a factor of f(x), if f(2) = 0

$f(2) = (2)^3 + (2)^2 - 4(2) - 4 \\[0.5em] = 8 + 4 - 8 - 4 \\[0.5em] = 0$

Hence, (x - 2) is a factor of x³ + x² - 4x - 4.

Now, factorizing x³ + x² - 4x - 4,

$\Rightarrow x^2(x + 1) - 4(x + 1) \\[0.5em] \Rightarrow (x^2 - 4)(x + 1) \\[0.5em] \Rightarrow (x - 2)(x + 2)(x + 1)$

Hence, x³ + x² - 4x - 4 = (x - 2)(x + 1)(x + 2).