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To factor the quadratic expression \( x^2 - 2x - 80 \), let's find two numbers that multiply to \(-80\) and add up to \(-2\).

1. **Identify factors of -80**:
   We need factors that add up to \(-2\). After testing different pairs, we find:
   \[
   -10 \text{ and } 8
   \]
   because \((-10) + 8 = -2\) and \((-10) \times 8 = -80\).

2. **Rewrite the expression**:
   Using these factors, we can rewrite the expression as:
   \[
   x^2 - 2x - 80 = (x - 10)(x + 8)
   \]

3. **Answer**:
   The correct answer is:
   \[
   (x - 10)(x + 8)
   \]

So, the answer is the third option: \((x - 10)(x + 8)\).

Would you like a more detailed breakdown of each step or have any questions?

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Here are some related questions to expand your understanding:

1. How do you factor a quadratic equation if the leading coefficient is not 1?
2. What is the difference between factoring and solving a quadratic equation?
3. How can you verify that your factorization is correct?
4. What if the equation had complex roots? How would factoring change?
5. How does completing the square compare to factoring in solving quadratics?

**Tip:** When factoring, always check your factors by expanding them to ensure they match the original expression.

Learn how to factor the quadratic expression x^2 - 2x - 80 completely. This problem demonstrates factoring techniques for quadratics by finding pairs of numbers that satisfy specific multiplication and addition conditions. Suitable for students in Grades 8-10.

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