continue with the exact expansions and finding the numerical solutions

Let's continue with the detailed expansion of both sides and find the numerical solutions for \(x\).

### Step-by-Step Expansion:

**Left Side:**
1. We already expanded \( (x - 2)(x + 2) \) to get \( x^2 - 4 \).
2. Substituting, we have:

   \[
   (x^2 - 4 - 2x) = x^2 - 2x - 4
   \]

3. Multiply by \( 3(x - 1)^2 \):

   Expanding \((x - 1)^2 = x^2 - 2x + 1\), the left side becomes:

   \[
   3(x^2 - 2x + 1) \cdot (x^2 - 2x - 4)
   \]

   Now, let's expand this product:

   \[
   = 3 \left[(x^2 - 2x)(x^2 - 2x - 4) + (x^2 - 2x - 4)\right]
   \]

   Continuing:

   \[
   = 3 \left[x^4 - 2x^3 - 4x^2 - 2x^3 + 4x^2 + 8x + x^2 - 2x - 4\right]
   \]

   Simplify:

   \[
   = 3 \left[x^4 - 4x^3 + x^2 + 6x - 4\right]
   \]

   Expand fully:

   \[
   = 3x^4 - 12x^3 + 3x^2 + 18x - 12
   \]

**Right Side:**
1. Previously simplified as:

   \[
   9x^2 - 24x + 12
   \]

### Equate and Simplify Further:

Now equate both sides:

\[
3x^4 - 12x^3 + 3x^2 + 18x - 12 = 9x^2 - 24x + 12
\]

Simplify by bringing all terms to one side:

\[
3x^4 - 12x^3 + 3x^2 - 9x^2 + 18x + 24x - 12 - 12 = 0
\]

\[
3x^4 - 12x^3 - 6x^2 + 42x - 24 = 0
\]

Divide by 3 for simplicity:

\[
x^4 - 4x^3 - 2x^2 + 14x - 8 = 0
\]

### Numerical Solutions:

To find the numerical solutions, we can solve this quartic equation:

\[
x^4 - 4x^3 - 2x^2 + 14x - 8 = 0
\]

Let's find the roots of this equation using numerical methods.The exact symbolic solutions for the quartic equation \(x^4 - 4x^3 - 2x^2 + 14x - 8 = 0\) are quite complex and involve nested square roots and imaginary components. For practical purposes, let’s find approximate numerical solutions:

Let me calculate the approximate numerical values of these roots.The approximate numerical solutions for the equation \(x^4 - 4x^3 - 2x^2 + 14x - 8 = 0\) are:

1. \(x \approx -1.927\) 
2. \(x \approx 0.749\)
3. \(x \approx 1.511\)
4. \(x \approx 3.666\)

These roots are real values, as the imaginary parts are extremely close to zero (on the order of \(10^{-30}\)) due to numerical approximations.

Would you like further details or have any questions about these solutions?

**Follow-Up Questions:**

1. Would you like to verify these solutions by substituting them back into the original equation?
2. Are you interested in exploring the behavior of the function graphically?
3. Do you want a deeper explanation of the steps involved in expanding the expressions?
4. Would you like to know more about the nature of quartic equations and their solutions?
5. Are there any specific roots or values you want to explore further?

**Tip:** When solving complex polynomial equations, it can be helpful to approximate numerical solutions to better understand their behavior, especially when symbolic solutions are cumbersome.

Explore the numerical solutions for the quartic equation x^4 - 4x^3 - 2x^2 + 14x - 8 with detailed step-by-step calculations. Understand how to find real roots using numerical methods. Suitable for advanced high school students and beyond.

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