To solve the equation $3x^3 + x^2 - 12x - 4 = 0$ by factoring using grouping, follow these steps:

1. Group the terms:

Group the first two and last two terms: $(3x^3 + x^2) - (12x + 4) = 0$

2. Factor out the greatest common factor (GCF) from each group:

• From $3x^3 + x^2$, factor out $x^2$: $x^2(3x + 1)$
• From $-12x - 4$, factor out $-4$: $-4(3x + 1)$

Now the equation looks like this: $x^2(3x + 1) - 4(3x + 1) = 0$

3. Factor out the common binomial factor $(3x + 1)$:

$(3x + 1)(x^2 - 4) = 0$

4. Factor the difference of squares $x^2 - 4$:

$(3x + 1)(x - 2)(x + 2) = 0$

5. Solve for $x$:

Set each factor equal to zero:

• $3x + 1 = 0$ gives $x = -\frac{1}{3}$
• $x - 2 = 0$ gives $x = 2$
• $x + 2 = 0$ gives $x = -2$

Final Solution:

The solutions are: $x = -\frac{1}{3}, \, x = 2, \, x = -2$

Would you like further details or have any questions?

---

Here are five related questions to expand on this topic:

1. How do you recognize when factoring by grouping is appropriate in a polynomial?
2. Can all cubic polynomials be factored using grouping? Why or why not?
3. What is the method for solving a cubic equation if factoring by grouping does not work?
4. What is the significance of factoring in solving polynomial equations?
5. How do you solve a cubic equation graphically, and how does it relate to factoring?

Tip: When factoring by grouping, always check if you can pull out a common binomial from both groups of terms. This is key to simplifying the equation.