Let's expand the expression \((x^2 - 3x - 2) \cdot (2x^2 - 4x - 8)\) using the distributive property (also known as FOIL for binomials). We will multiply each term in the first polynomial by each term in the second polynomial.

\[
(x^2 - 3x - 2) \cdot (2x^2 - 4x - 8)
\]

### Step-by-step solution:

1. Multiply each term in the first expression by \(2x^2\):
   \[
   x^2 \cdot 2x^2 = 2x^4
   \]
   \[
   -3x \cdot 2x^2 = -6x^3
   \]
   \[
   -2 \cdot 2x^2 = -4x^2
   \]

2. Multiply each term in the first expression by \(-4x\):
   \[
   x^2 \cdot (-4x) = -4x^3
   \]
   \[
   -3x \cdot (-4x) = 12x^2
   \]
   \[
   -2 \cdot (-4x) = 8x
   \]

3. Multiply each term in the first expression by \(-8\):
   \[
   x^2 \cdot (-8) = -8x^2
   \]
   \[
   -3x \cdot (-8) = 24x
   \]
   \[
   -2 \cdot (-8) = 16
   \]

### Now, combine all the terms:
\[
2x^4 - 6x^3 - 4x^2 - 4x^3 + 12x^2 + 8x - 8x^2 + 24x + 16
\]

### Combine like terms:
- \(2x^4\)
- \(-6x^3 - 4x^3 = -10x^3\)
- \(-4x^2 + 12x^2 - 8x^2 = 0x^2\)
- \(8x + 24x = 32x\)
- \(16\) (constant term)

### Final simplified expression:
\[
2x^4 - 10x^3 + 32x + 16
\]

This is the expanded and simplified result.

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Do you want more details or have any questions?

### Related questions:
1. How do you factor polynomial expressions?
2. What is the distributive property in algebra?
3. How do you find the roots of a quadratic equation?
4. What is the difference between expanding and factoring polynomials?
5. How do you solve cubic equations?

### Tip:
When expanding polynomials, be sure to organize the terms clearly to avoid missing any like terms during the combination step.

Expand and simplify the polynomial expression (x² - 3x - 2) • (2x² - 4x - 8) using the distributive property. This step-by-step solution covers key algebraic concepts like polynomial expansion and combining like terms. Suitable for students in Grades 10-12.

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