The image shows a problem where you need to evaluate and simplify the difference quotient for the function $f(x) = 3x^2 + 4x$. The expression given is:

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

Let's work through this step by step.

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

Given $f(x) = 3x^2 + 4x$, substitute $x+h$ into the function:

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

Step 2: Expand and simplify $f(x+h)$

Expand $(x+h)^2$:

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

Now distribute the constants:

$f(x+h) = 3x^2 + 6xh + 3h^2 + 4x + 4h$

Step 3: Subtract $f(x)$

Now, subtract $f(x) = 3x^2 + 4x$ from $f(x+h)$:

$f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 + 4x + 4h) - (3x^2 + 4x)$

Simplify by canceling out the like terms $3x^2$ and $4x$:

$f(x+h) - f(x) = 6xh + 3h^2 + 4h$

Step 4: Divide by $h$

Finally, divide the entire expression by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 + 4h}{h}$

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

$\frac{f(x+h) - f(x)}{h} = 6x + 3h + 4$

Final Answer

The simplified expression for the difference quotient is:

$\boxed{6x + 3h + 4}$

Would you like any more details, or do you have any questions?

Here are some related questions you might explore:

1. What is the geometric interpretation of the difference quotient?
2. How does the difference quotient relate to the derivative?
3. Can the difference quotient be used for functions other than polynomials?
4. What happens to the difference quotient as $h$ approaches zero?
5. How do you find the derivative of a quadratic function using first principles?

Tip: Understanding how to manipulate and simplify expressions algebraically is crucial for working with calculus concepts like derivatives!