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The problem now asks to rewrite the expression \(\frac{\alpha^4 + \beta^4}{\alpha \beta}\) using \((\alpha + \beta)\) and \((\alpha \beta)\).

Let's start by expressing \(\alpha^4 + \beta^4\) in terms of \((\alpha + \beta)\) and \((\alpha \beta)\).

### Step-by-Step Solution:

1. **Square the sum of squares**:  
   We can express \(\alpha^4 + \beta^4\) as \((\alpha^2 + \beta^2)^2 - 2(\alpha \beta)^2\).

2. **Express \(\alpha^2 + \beta^2\) using \((\alpha + \beta)\) and \((\alpha \beta)\)**:  
   The identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta\) can be used.

   So, \[
   \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta
   \]

3. **Substitute back into the expression for \(\alpha^4 + \beta^4\)**:  
   \[
   \alpha^4 + \beta^4 = ((\alpha + \beta)^2 - 2\alpha \beta)^2 - 2(\alpha \beta)^2
   \]

4. **Expand and simplify**:  
   \[
   (\alpha + \beta)^4 - 4(\alpha + \beta)^2(\alpha \beta) + 4(\alpha \beta)^2 - 2(\alpha \beta)^2
   \]

   Combine like terms:
   \[
   \alpha^4 + \beta^4 = (\alpha + \beta)^4 - 4(\alpha + \beta)^2(\alpha \beta) + 2(\alpha \beta)^2
   \]

5. **Rewrite the original fraction**:  
   \[
   \frac{\alpha^4 + \beta^4}{\alpha \beta} = \frac{(\alpha + \beta)^4 - 4(\alpha + \beta)^2(\alpha \beta) + 2(\alpha \beta)^2}{\alpha \beta}
   \]

6. **Divide each term by \(\alpha \beta\)**:  
   \[
   \frac{(\alpha + \beta)^4}{\alpha \beta} - 4\frac{(\alpha + \beta)^2(\alpha \beta)}{\alpha \beta} + 2\frac{(\alpha \beta)^2}{\alpha \beta}
   \]

   Simplifying each term:
   \[
   = \frac{(\alpha + \beta)^4}{\alpha \beta} - 4(\alpha + \beta)^2 + 2\alpha \beta
   \]

### Final Expression:

Thus, the expression \(\frac{\alpha^4 + \beta^4}{\alpha \beta}\) written in terms of \((\alpha + \beta)\) and \((\alpha \beta)\) is:

\[
\frac{(\alpha + \beta)^4}{\alpha \beta} - 4(\alpha + \beta)^2 + 2\alpha \beta
\]

Would you like more details or have any questions?

Here are some questions to further explore this topic:

1. How would you generalize the expression for \(\frac{\alpha^n + \beta^n}{\alpha \beta}\) for any \(n\)?
2. What happens to the expression if \(\alpha = \beta\)?
3. Can you derive a similar expression for \(\alpha^5 + \beta^5\) using \((\alpha + \beta)\) and \((\alpha \beta)\)?
4. What other polynomial identities can be helpful in problems like this?
5. How can these identities help in solving polynomial equations?

**Tip:** Always look for patterns in polynomial expressions. Understanding these patterns can simplify your work with algebraic identities.

Explore how to rewrite the expression (α^4 + β^4) / (αβ) using the identities involving (α + β) and (αβ). This detailed solution covers polynomial identities and algebraic manipulations, suitable for advanced high school students.

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