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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.

Find the difference quotient of f(x) = 2x^2 + x + 3. Simplify your answer.

Learn how to compute the difference quotient of the quadratic function f(x) = 2x^2 + x + 3. This step-by-step solution simplifies the expression and explains the underlying algebraic concepts. Suitable for high school calculus students.

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