Solve.​ (Hint: factor by​ grouping.)
3 x cubed plus x squared minus 12 x minus 4 equals 03x3+x2−12x−4=0

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?

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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.

Solve by factoring using grouping: 3x^3 + x^2 - 12x - 4 = 0

This solution demonstrates how to solve the cubic polynomial equation 3x^3 + x^2 - 12x - 4 = 0 by factoring using grouping. Key algebraic concepts include factoring by grouping and the difference of squares formula. The problem is suitable for students in Grades 10-12 and covers advanced algebraic techniques.

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