To find and simplify the difference quotient $\frac{f(3+h) - f(3)}{h}$ for the function $f(x) = x^2 - 2x + 2$, we will follow these steps:

1. Calculate $f(3)$: $f(3) = 3^2 - 2 \cdot 3 + 2 = 9 - 6 + 2 = 5$

2. Calculate $f(3+h)$: $f(3+h) = (3+h)^2 - 2(3+h) + 2$ Expand and simplify: $f(3+h) = (3+h)(3+h) - 2(3+h) + 2 = 9 + 6h + h^2 - 6 - 2h + 2 = h^2 + 4h + 5$

3. Form the difference quotient $\frac{f(3+h) - f(3)}{h}$: $\frac{f(3+h) - f(3)}{h} = \frac{(h^2 + 4h + 5) - 5}{h} = \frac{h^2 + 4h}{h}$

4. Simplify the expression: $\frac{h^2 + 4h}{h} = h + 4$

So, the simplified form of the difference quotient is: $\boxed{h + 4}$

Would you like more details or have any questions about this process?

Here are 5 relative questions to further your understanding:

1. How would the difference quotient change if the function was $f(x) = x^2 - 2x + 3$ instead?
2. What is the geometric interpretation of the difference quotient?
3. How does the difference quotient relate to the derivative of the function at $x = 3$?
4. Can you compute the difference quotient for $f(x) = x^3 - x + 1$ at $x = 2$?
5. What happens to the difference quotient as $h$ approaches zero?

Tip: The difference quotient is a fundamental concept that leads to the derivative, which measures how a function changes as its input changes.