Let's solve the problem step by step.

Given:

The function is $f(x) = 2x^2 - 5x + 8$.

Part (A) $f(x + h)$:

To find $f(x + h)$, substitute $x + h$ for $x$ in the function $f(x)$:

$f(x + h) = 2(x + h)^2 - 5(x + h) + 8$

Now, expand the expression:

1. Expand $(x + h)^2$: $(x + h)^2 = x^2 + 2xh + h^2$

2. Substitute this into the function: $f(x + h) = 2(x^2 + 2xh + h^2) - 5(x + h) + 8$

3. Distribute the constants: $f(x + h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 8$

So, the simplified expression for $f(x + h)$ is:

$\boxed{f(x + h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 8}$

Part (B) $f(x + h) - f(x)$:

Next, subtract $f(x)$ from $f(x + h)$:

$f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 - 5x - 5h + 8\right) - \left(2x^2 - 5x + 8\right)$

Now, simplify the expression:

1. Combine like terms: $f(x + h) - f(x) = 4xh + 2h^2 - 5h$

So, the simplified expression for $f(x + h) - f(x)$ is:

$\boxed{f(x + h) - f(x) = 4xh + 2h^2 - 5h}$

Part (C) $\frac{f(x + h) - f(x)}{h}$:

Finally, divide the result from part (B) by $h$:

$\frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 - 5h}{h}$

Simplify the expression by dividing each term by $h$:

$\frac{f(x + h) - f(x)}{h} = 4x + 2h - 5$

So, the simplified expression for $\frac{f(x + h) - f(x)}{h}$ is:

$\boxed{\frac{f(x + h) - f(x)}{h} = 4x + 2h - 5}$

---

Would you like further details or have any questions?

Here are some related questions:

1. How do you find the derivative of a function using the difference quotient?
2. What is the significance of the term "h" approaching zero in calculus?
3. How would you find the derivative of $f(x) = 2x^2 - 5x + 8$ using the power rule?
4. Can you explain why the function simplifies in each step?
5. What are some real-world applications of derivatives?

Tip: When simplifying expressions, always distribute and combine like terms carefully to avoid errors.