To write the polynomial $x^3 + x^2 - 8x - 12$ as a product of linear factors, we need to factor it fully. Let's proceed step by step:

Step 1: Look for possible rational roots.

Using the Rational Root Theorem, the possible rational roots are factors of the constant term (-12) divided by the factors of the leading coefficient (1), so the possible roots are:

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

Step 2: Test the possible roots using synthetic division or substitution.

Let’s try $x = 2$:

$f(2) = 2^3 + 2^2 - 8(2) - 12 = 8 + 4 - 16 - 12 = -16$

$x = 2$ is not a root.

Now, try $x = -2$:

$f(-2) = (-2)^3 + (-2)^2 - 8(-2) - 12 = -8 + 4 + 16 - 12 = 0$

So, $x = -2$ is a root.

Step 3: Perform synthetic division.

We will now divide $x^3 + x^2 - 8x - 12$ by $x + 2$ using synthetic division: