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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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.