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Let's solve the function \( f(x) = \frac{5x^2 - 8x}{2x} \).

### Step-by-Step Solution

1. **Simplify the expression**:  
   First, let's simplify the given function by dividing both the numerator and the denominator by \(x\). However, remember that \(x \neq 0\) because we can't divide by zero.

   \[
   f(x) = \frac{5x^2 - 8x}{2x}
   \]

2. **Factor out \(x\) from the numerator**:
   \[
   f(x) = \frac{x(5x - 8)}{2x}
   \]

3. **Cancel out the common \(x\) terms**:
   Since \(x \neq 0\), we can divide both the numerator and the denominator by \(x\):

   \[
   f(x) = \frac{5x - 8}{2}
   \]

4. **Final Simplified Form**:
   \[
   f(x) = \frac{5x}{2} - 4
   \]

### Conclusion

The simplified form of the function \( f(x) = \frac{5x^2 - 8x}{2x} \) is \( f(x) = \frac{5x}{2} - 4 \), where \( x \neq 0 \).

Would you like more details on this solution, or do you have any further questions? Here are some related questions you might find interesting:

1. How would the function change if the denominator was different, like \(2x^2\)?
2. What is the domain of this function?
3. How do you find the derivative of the simplified function?
4. What are the critical points of the simplified function?
5. How would you graph the simplified function?

**Tip:** Always check for restrictions in the domain when simplifying rational functions, especially when variables are in the denominator.

Learn how to simplify the rational function f(x) = (5x^2 - 8x) / (2x) with a detailed step-by-step solution. This problem covers key algebraic concepts, including rational functions and simplification techniques. Suitable for students in Grades 10-12.

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