Let's work through this problem step-by-step.

The given function is:

$f(x) = 2x^2 + x + 3$

We are tasked with finding the difference quotient:

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

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

We need to plug $x+h$ into the function $f(x)$:

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

Expanding each term:

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

Step 2: Subtract $f(x)$ from $f(x+h)$

We already have:

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

And $f(x) = 2x^2 + x + 3$. Now subtract $f(x)$ from $f(x+h)$:

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

Simplifying:

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

Step 3: Divide by $h$

Now, divide the result by $h$:

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

Simplifying:

$= 4x + 2h + 1$

Final Answer:

$\frac{f(x+h) - f(x)}{h} = 4x + 2h + 1$

Would you like a more detailed explanation or have any questions?

Here are five related questions:

1. What happens to the difference quotient as $h \to 0$?
2. Can you compute the derivative of $f(x) = 2x^2 + x + 3$ using the difference quotient method?
3. How does the difference quotient relate to the concept of a derivative?
4. What would the difference quotient look like for a linear function like $f(x) = 3x + 5$?
5. Can you apply the difference quotient to a cubic function like $f(x) = x^3$?

Tip: When solving difference quotients, always expand and simplify expressions carefully to avoid errors.