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The question asks to find the difference quotient for the function \( f(x) = x^2 + 6 \), which is given by the formula:

\[
\frac{f(x+h) - f(x)}{h}
\]

### Step-by-step solution:

1. **Find \( f(x+h) \):**
   \[
   f(x+h) = (x+h)^2 + 6
   \]
   Expand \( (x+h)^2 \):
   \[
   (x+h)^2 = x^2 + 2xh + h^2
   \]
   Therefore:
   \[
   f(x+h) = x^2 + 2xh + h^2 + 6
   \]

2. **Find the difference \( f(x+h) - f(x) \):**
   Since \( f(x) = x^2 + 6 \), we compute:
   \[
   f(x+h) - f(x) = (x^2 + 2xh + h^2 + 6) - (x^2 + 6)
   \]
   Simplifying:
   \[
   f(x+h) - f(x) = 2xh + h^2
   \]

3. **Divide by \( h \) to get the difference quotient:**
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2}{h}
   \]
   Simplifying:
   \[
   \frac{f(x+h) - f(x)}{h} = 2x + h
   \]

Thus, the difference quotient is:

\[
\boxed{2x + h}
\]

Would you like more details on any part of the solution? Here are 5 related questions:

1. How does the difference quotient relate to the derivative?
2. What happens to the difference quotient as \( h \to 0 \)?
3. Can the difference quotient be used to approximate slopes of curves?
4. How do we interpret \( f(x+h) - f(x) \) geometrically?
5. Can you explain the significance of \( h \neq 0 \)?

**Tip:** The difference quotient is a fundamental concept in calculus, used to compute the derivative of a function as \( h \) approaches zero.

For the function f(x) = x^2 + 6, find the difference quotient (f(x+h) - f(x)) / h for h ≠ 0.

Learn how to calculate the difference quotient for the quadratic function f(x) = x^2 + 6. This detailed solution covers essential calculus concepts and provides a step-by-step guide to finding the difference quotient, which is a foundation for understanding derivatives.

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