Prove that (5x + 4) is a factor of 5x³ + 4x² - 5x - 4. Hence, factorise the given polynomial completely.

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

f(x) = 5x³ + 4x² - 5x - 4

Given, (5x + 4) or 5(x - (-$\dfrac{4}{5}$)) is a factor of f(x) hence, let's find f(-$\dfrac{4}{5}$).

$\therefore f(-\dfrac{4}{5}) = 5\big(-\dfrac{4}{5}\big)^3 + 4\big(-\dfrac{4}{5}\big)^2 -5\big(-\dfrac{4}{5}\big) - 4 \\[1em] = 5\big(-\dfrac{64}{125}\big) + 4\big(\dfrac{16}{25}\big) + 4 - 4 \\[1em] = -\dfrac{64}{25} + \dfrac{64}{25} \\[1em] = 0$

Since, f(-$\dfrac{4}{5}$) = 0, hence (5x + 4) is a factor of f(x).

Now, factorising the equation 5x³ + 4x² - 5x - 4

$\Rightarrow x^2(5x + 4) - 1(5x + 4) \\[0.5em] \Rightarrow (x^2 - 1)(5x + 4) \\[0.5em] \Rightarrow (x^2 - (1)^2)(5x + 4) \\[0.5em] \Rightarrow (x - 1)(x + 1)(5x + 4)$

Hence, 5x³ + 4x² - 5x - 4 = (x - 1)(x + 1)(5x + 4).