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Let's analyze the given information and solve for the required value.

### Given:
1. The roots of the quadratic equation \( x^2 + 3x - 5 = 0 \) are \( \alpha \) and \( \beta \).
2. \( \alpha^2 + \beta^2 = 19 \)
3. \( \alpha - \beta = \sqrt{29} \)
4. \( \alpha > \beta \)
5. We need to find the exact value of \( \alpha^4 - \beta^4 \).

### Steps to Solve:

1. **Sum and Product of Roots:**
   For the quadratic equation \( x^2 + 3x - 5 = 0 \), the sum of the roots \( \alpha + \beta = -\text{coefficient of } x = -3 \), and the product of the roots \( \alpha \beta = \text{constant term} = -5 \).

2. **Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):**
   \[
   \alpha^2 + \beta^2 = (-3)^2 - 2(-5) = 9 + 10 = 19
   \]
   This matches the given information.

3. **Finding \( \alpha^2 - \beta^2 \):**
   Using the identity \( \alpha^2 - \beta^2 = (\alpha - \beta)(\alpha + \beta) \):
   \[
   \alpha^2 - \beta^2 = \sqrt{29} \times (-3) = -3\sqrt{29}
   \]

4. **Finding \( \alpha^4 - \beta^4 \):**
   Using the identity \( \alpha^4 - \beta^4 = (\alpha^2 - \beta^2)(\alpha^2 + \beta^2) \):
   \[
   \alpha^4 - \beta^4 = (-3\sqrt{29}) \times 19 = -57\sqrt{29}
   \]

So, the exact value of \( \alpha^4 - \beta^4 \) is \( \boxed{-57\sqrt{29}} \).

Do you have any questions or need further details on any part of the solution?

### Follow-up Questions:
1. How can we find the roots \( \alpha \) and \( \beta \) individually from the quadratic equation?
2. What are the steps to verify if \( \alpha \) and \( \beta \) satisfy the quadratic equation?
3. How would the solution change if \( \alpha < \beta \)?
4. Can you derive the value of \( \alpha^4 + \beta^4 \) using similar identities?
5. What is the importance of sum and product of roots in solving quadratic equations?

### Tip:
When solving quadratic equations, always consider using identities involving the sum and product of roots to simplify complex expressions like \( \alpha^4 - \beta^4 \).

The roots of the equation x^2 + 3x - 5 = 0 are α and β. Given: α^2 + β^2 = 19, α - β = √29, α > β, find the exact value of α^4 - β^4.

Learn how to find the exact value of α^4 - β^4 for the roots of the quadratic equation x^2 + 3x - 5 = 0. This problem involves key algebraic concepts like sum and product of roots, difference of squares, and polynomial identities. Suitable for students in Grades 9-11.

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