You have provided an image containing the function $f(x) = 2 + 2x^2$ and you're asking to calculate the difference quotient:

$\frac{f(a+h) - f(a)}{h}$

Let's solve this step-by-step:

1. Substitute $f(x)$ into the difference quotient. First, we need to find $f(a+h)$ and $f(a)$.

$f(a+h) = 2 + 2(a+h)^2$ $f(a) = 2 + 2a^2$

2. Expand $f(a+h)$.

$f(a+h) = 2 + 2(a^2 + 2ah + h^2) = 2 + 2a^2 + 4ah + 2h^2$

3. Now, subtract $f(a)$ from $f(a+h)$:

$f(a+h) - f(a) = \left( 2 + 2a^2 + 4ah + 2h^2 \right) - \left( 2 + 2a^2 \right)$ $f(a+h) - f(a) = 4ah + 2h^2$

4. Divide by $h$:

$\frac{f(a+h) - f(a)}{h} = \frac{4ah + 2h^2}{h}$

5. Simplify the expression:

$\frac{f(a+h) - f(a)}{h} = 4a + 2h$

Thus, the simplified form of the difference quotient is:

$\frac{f(a+h) - f(a)}{h} = 4a + 2h$

If you have any questions or need further clarification, feel free to ask!

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Here are five related questions to deepen understanding:

1. What is the difference quotient used for in calculus?
2. How would the derivative of $f(x) = 2 + 2x^2$ relate to this expression?
3. What happens to the difference quotient as $h \to 0$?
4. Can this technique be applied to functions with more complex expressions, like polynomials of higher degrees?
5. What is the geometric interpretation of the difference quotient?

Tip: The difference quotient is the foundation of the derivative, which gives the slope of the tangent line to the function at any point!