To solve the given expression:

$\frac{f(x+h) - f(x)}{h} = x^2 + 2xh + h^2 + x + h + 3 - (x^2 + x + 3)$

First, simplify the numerator:

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

Now, distribute and combine like terms:

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

The $x^2$ and $-x^2$ terms cancel out, as do the $x$ and $-x$ terms, and the $3$ and $-3$ terms:

$f(x+h) - f(x) = 2xh + h^2 + h$

Next, divide the entire expression by $h$:

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

Separate the terms in the numerator:

$\frac{2xh}{h} + \frac{h^2}{h} + \frac{h}{h}$

Simplify each term:

$= 2x + h + 1$

Therefore, we have:

$\frac{f(x+h) - f(x)}{h} = 2x + h + 1$

This simplification shows the expression in its final form.

Do you want more details or have any questions?

Here are some related questions you might find interesting:

1. How does the limit of $\frac{f(x+h) - f(x)}{h}$ as $h$ approaches 0 relate to the derivative of $f(x)$?
2. What is the derivative of $f(x) = x^2 + x + 3$?
3. How do you find the derivative of a polynomial function?
4. What role does the term $h^2$ play in the limit definition of a derivative?
5. How can you interpret the term $2xh$ geometrically?
6. Why do the constant terms cancel out when simplifying $f(x+h) - f(x)$?
7. How does the expression change if $f(x)$ includes higher degree terms like $x^3$?
8. Can you generalize the result for any quadratic polynomial function?

Tip: When finding the limit as $h$ approaches 0 for $\frac{f(x+h) - f(x)}{h}$, always simplify the numerator first to make it easier to divide by $h$.