To find $\frac{f(x+h) - f(x)}{h}$, given $f(x) = 2x^2 + 5x - 3$, follow these steps:

Step 1: Find $f(x + h)$

We substitute $x + h$ into the function $f(x)$:

$f(x + h) = 2(x + h)^2 + 5(x + h) - 3$

Expand the terms:

$= 2(x^2 + 2xh + h^2) + 5(x + h) - 3$ $= 2x^2 + 4xh + 2h^2 + 5x + 5h - 3$

Step 2: Calculate $f(x+h) - f(x)$

Now subtract $f(x)$ from $f(x + h)$:

$f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 + 5x + 5h - 3\right) - \left(2x^2 + 5x - 3\right)$

Simplifying this:

$= 2x^2 + 4xh + 2h^2 + 5x + 5h - 3 - 2x^2 - 5x + 3$ $= 4xh + 2h^2 + 5h$

Step 3: Divide by $h$

Now, divide the expression by $h$:

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

Simplifying:

$= 4x + 2h + 5$

Final Answer:

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

Let me know if you need further clarification or details.

Here are 5 follow-up questions related to this:

1. What happens to the expression as $h \to 0$?
2. How would you compute the derivative of $f(x)$ using the limit definition?
3. How does this result relate to the concept of the difference quotient?
4. Can you find $f'(x)$ using another method like power rule?
5. How would the expression change if $f(x)$ were a cubic function?

Tip: The expression $\frac{f(x+h) - f(x)}{h}$ is often called the difference quotient, and it's a foundational concept for finding derivatives.