To find and simplify the difference quotient for the function $f(x) = 2x^2 + 6x + 3$, we will follow these steps:

Step 1: Write the formula for the difference quotient

The difference quotient is given by:

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

Step 2: Compute $f(x+h)$

Substitute $x + h$ into the function $f(x)$:

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

Expanding this:

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

Simplify further:

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

Step 3: Compute $f(x+h) - f(x)$

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

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

Simplify by canceling out the like terms:

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

Step 4: Divide by $h$

Now, divide the result by $h$:

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

Simplify by factoring out $h$ from the numerator:

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

Final Answer

The simplified form of the difference quotient is:

$4x + 2h + 6$

This is the simplified expression for the difference quotient of the given function.

Would you like more details or have any questions?

Here are five related questions:

1. What does the difference quotient represent geometrically?
2. How would the difference quotient change if the function were cubic instead of quadratic?
3. What happens to the difference quotient as $h$ approaches zero?
4. Can you apply the difference quotient to a trigonometric function like $\sin(x)$?
5. How does the difference quotient relate to the derivative of a function?

Tip: The difference quotient is a fundamental concept in calculus, as it provides the basis for defining the derivative, which measures the rate of change of a function.