To simplify the expression \((x^2 - 3x - 2) \times (2x^2 - 4x - 8)\), we can use the distributive property (also known as FOIL in the case of binomials). Here's the step-by-step process:

### Step 1: Factor both polynomials
- \((x^2 - 3x - 2)\) factors as \((x - 2)(x + 1)\)
- \((2x^2 - 4x - 8)\) factors as \(2(x - 2)(x + 2)\)

### Step 2: Multiply the factored forms
\[
(x - 2)(x + 1) \times 2(x - 2)(x + 2)
\]

### Step 3: Combine the common terms
We have \((x - 2)\) in both expressions, so it simplifies to:
\[
(x - 2)^2 \times 2(x + 1)(x + 2)
\]

### Step 4: Expand the expression
Now expand \((x - 2)^2 = (x^2 - 4x + 4)\) and multiply by \(2(x + 1)(x + 2)\).

- First expand \((x + 1)(x + 2)\):
\[
(x + 1)(x + 2) = x^2 + 3x + 2
\]

- Now multiply everything:
\[
2(x^2 - 4x + 4)(x^2 + 3x + 2)
\]

This is the fully expanded form of the product.

Would you like me to continue simplifying, or do you need more details on any specific step?

Here are five questions to extend this understanding:

1. How do you factor a quadratic trinomial in general?
2. What is the distributive property, and how is it applied here?
3. How do you simplify \((x + a)(x + b)\)?
4. How would the expression change if \(2x^2 - 4x - 8\) had different signs?
5. What are the advantages of factoring before expanding?

**Tip:** Always check if you can factor terms before multiplying polynomials—it can save a lot of time in simplifying.

Explore the step-by-step solution to simplify the expression (x² - 3x - 2) x (2x² - 4x - 8). This solution covers key algebraic concepts such as factoring, polynomial multiplication, and the distributive property. Suitable for students in Grades 8-10.

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