The problem asks to factor the cubic polynomial $x^3 + 2x^2 + 6x + 12$ completely.

1. Problem Description

The problem asks to factor the cubic polynomial $x^3 + 2x^2 + 6x + 12$ completely.

2. Solution Steps

We can try factoring by grouping.

$x^3 + 2x^2 + 6x + 12 = (x^3 + 2x^2) + (6x + 12)$.

Now we factor out the greatest common factor (GCF) from each group:

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

$6x + 12 = 6(x + 2)$

Now we have:

$x^2(x + 2) + 6(x + 2)$.

We can factor out the common factor $(x + 2)$:

$(x + 2)(x^2 + 6)$.

The term $x^2 + 6$ cannot be factored further using real numbers because it has no real roots.

3. Final Answer

$(x+2)(x^2+6)$