The problem is to factor the quadratic expression $3x^2 - 4x - 4$ completely.

1. Problem Description

The problem is to factor the quadratic expression $3x^2 - 4x - 4$ completely.

2. Solution Steps

We need to find two binomials such that their product is equal to $3x^2 - 4x - 4$. We are looking for two numbers whose product is $3 \cdot (-4) = -12$ and whose sum is $-4$. The two numbers are $-6$ and $2$, since $(-6)(2) = -12$ and $-6 + 2 = -4$.

Now, we rewrite the middle term $-4x$ as $-6x + 2x$:

$3x^2 - 4x - 4 = 3x^2 - 6x + 2x - 4$

Next, we factor by grouping:

$3x^2 - 6x + 2x - 4 = 3x(x - 2) + 2(x - 2)$

Now, we factor out the common binomial factor $(x - 2)$:

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

3. Final Answer

The factored form of the expression $3x^2 - 4x - 4$ is $(3x + 2)(x - 2)$.