To factor the quadratic expression $3x^2 - 4x - 4$, we need to find two binomials whose product gives us the original expression. Let's factor it step-by-step:

Step 1: Multiply the leading coefficient (3) by the constant term (-4).

This gives $3 \times (-4) = -12$.

Step 2: Find two numbers that multiply to -12 and add to -4 (the middle coefficient).

The two numbers that work are -6 and +2, because: $-6 \times 2 = -12 \quad \text{and} \quad -6 + 2 = -4.$

Step 3: Rewrite the middle term using -6 and +2.

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

Step 4: Factor by grouping.

Group the terms: $(3x^2 - 6x) + (2x - 4).$ Factor out the greatest common factors from each group: $3x(x - 2) + 2(x - 2).$

Step 5: Factor out the common binomial factor.

Both terms contain $(x - 2)$, so factor that out: $(3x + 2)(x - 2).$

Thus, the factors of $3x^2 - 4x - 4$ are $(3x + 2)(x - 2)$.

Step 6: Conclusion

One of the factors is $x - 2$, so the correct answer from the choices is:

$\boxed{x - 2}.$

Would you like more details on any of the steps or have any additional questions?

Here are 5 related questions for further practice:

1. How do you factor quadratic expressions when the leading coefficient is greater than 1?
2. What is the general process for factoring by grouping?
3. How can you confirm that the factors are correct by multiplying them back?
4. How would you factor $4x^2 - 11x + 6$?
5. How does factoring help in solving quadratic equations?

Tip: Always double-check your factorization by expanding the factors to see if they give you the original quadratic expression.