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.