Let's solve the given equation step by step.

We have:

\[
2x(x^2 - 16) = 8(x^2 - 16 - 2x + 4)
\]

### Step 1: Simplify both sides

First, let's expand both sides of the equation.

#### Left-hand side:
\[
2x(x^2 - 16) = 2x \cdot x^2 - 2x \cdot 16 = 2x^3 - 32x
\]

#### Right-hand side:
\[
8(x^2 - 16 - 2x + 4) = 8(x^2 - 12 - 2x)
\]
Now distribute the 8:
\[
= 8x^2 - 96 - 16x
\]

So, the equation becomes:

\[
2x^3 - 32x = 8x^2 - 96 - 16x
\]

### Step 2: Bring all terms to one side

Let's bring all terms to one side of the equation to set it equal to zero.

\[
2x^3 - 32x - 8x^2 + 96 + 16x = 0
\]

Simplify:

\[
2x^3 - 8x^2 - 16x + 96 = 0
\]

### Step 3: Factor or use numerical methods

We now have a cubic equation:

\[
2x^3 - 8x^2 - 16x + 96 = 0
\]

We can factor out 2:

\[
2(x^3 - 4x^2 - 8x + 48) = 0
\]

Divide both sides by 2:

\[
x^3 - 4x^2 - 8x + 48 = 0
\]

This is a cubic equation, which may need numerical methods like synthetic division or using the Rational Root Theorem to find its solutions. 

Would you like to proceed with finding the roots or explore another method to solve it?

---

Here are five questions to expand on this topic:

1. What are the possible strategies for solving cubic equations?
2. Can synthetic division be used to simplify this problem further?
3. What are some real-world applications of cubic equations?
4. How can the Rational Root Theorem help in finding solutions to cubic equations?
5. How does the nature of the roots change based on the discriminant of a cubic equation?

**Tip:** When simplifying polynomial equations, always look for common factors that you can factor out to make the equation easier to solve.

Solve the cubic equation 2x(x^2 - 16) = 8(x^2 - 16 - 2x + 4) using step-by-step methods. Learn how to simplify, factor, and apply the Rational Root Theorem to find solutions. Ideal for students in Grades 10-12 exploring advanced algebra topics.

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