Factor the expression $x^3 - 8$ using the difference of cubes formula.

1. Problem Description

Factor the expression $x^3 - 8$ using the difference of cubes formula.

2. Solution Steps

The difference of cubes formula is:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

In this case, we have $x^3 - 8$, which can be written as $x^3 - 2^3$.

So, $a = x$ and $b = 2$.

Applying the formula, we get:

$x^3 - 2^3 = (x - 2)(x^2 + x(2) + 2^2)$

$x^3 - 8 = (x - 2)(x^2 + 2x + 4)$

The quadratic $x^2 + 2x + 4$ has discriminant $\Delta = 2^2 - 4(1)(4) = 4 - 16 = -12$, which is negative. Therefore, the quadratic is irreducible over the real numbers.

3. Final Answer

$(x - 2)(x^2 + 2x + 4)$