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The problem asks you to find and simplify the **difference quotient** for the function \( f(x) = 8x^3 + 5x \). The difference quotient is given by:

\[
\frac{f(x+h) - f(x)}{h}, \quad h \neq 0
\]

### Steps to solve:

1. **Substitute \(f(x)\)** into the difference quotient formula:
   \[
   f(x) = 8x^3 + 5x
   \]
   Now, compute \(f(x+h)\) by substituting \(x + h\) for \(x\) in \(f(x)\):
   \[
   f(x + h) = 8(x+h)^3 + 5(x+h)
   \]

2. **Expand \(f(x+h)\):**
   Expand \( (x+h)^3 \) first:
   \[
   (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
   \]
   So,
   \[
   f(x+h) = 8(x^3 + 3x^2h + 3xh^2 + h^3) + 5(x + h)
   \]
   Expanding this gives:
   \[
   f(x+h) = 8x^3 + 24x^2h + 24xh^2 + 8h^3 + 5x + 5h
   \]

3. **Set up the difference quotient:**
   Now compute \( f(x+h) - f(x) \):
   \[
   f(x+h) - f(x) = (8x^3 + 24x^2h + 24xh^2 + 8h^3 + 5x + 5h) - (8x^3 + 5x)
   \]
   Simplifying this expression:
   \[
   f(x+h) - f(x) = 24x^2h + 24xh^2 + 8h^3 + 5h
   \]

4. **Divide by \(h\):**
   Now, divide the result by \(h\):
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{24x^2h + 24xh^2 + 8h^3 + 5h}{h}
   \]
   Factor out \(h\) from the numerator:
   \[
   \frac{f(x+h) - f(x)}{h} = 24x^2 + 24xh + 8h^2 + 5
   \]

### Final Answer:
\[
\frac{f(x+h) - f(x)}{h} = 24x^2 + 24xh + 8h^2 + 5
\]

This is the simplified form of the difference quotient for the given function \( f(x) = 8x^3 + 5x \).

Would you like further details or have any questions?

### Related Questions:
1. What happens to the difference quotient as \( h \) approaches zero?
2. How does the difference quotient relate to the derivative of a function?
3. How do you calculate the derivative of \( f(x) = 8x^3 + 5x \) directly?
4. Can you explain how to expand \( (x+h)^3 \) step by step?
5. What is the geometric interpretation of the difference quotient?

### Tip:
The difference quotient is the foundation for understanding derivatives, as it represents the average rate of change of the function over an interval.

Find the difference quotient for the function f(x) = 8x^3 + 5x and simplify the answer.

This page provides a step-by-step solution for finding the difference quotient of the function f(x) = 8x^3 + 5x, which involves algebraic expansion and simplification techniques. Suitable for high school calculus students.

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