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We are tasked with determining the expression \(\frac{g(x+h) - g(x)}{h}\), where \(g(x)\) is given by:

\[
g(x) = -2x^2 + x + 6
\]

This expression is a difference quotient, which is used to find the slope of the secant line through two points on the graph of the function \(g(x)\), and is closely related to the derivative of \(g(x)\). Let's go through the steps to find the complete solution.

### Step 1: Calculate \(g(x + h)\)
Substitute \(x + h\) into the function \(g(x)\) to find \(g(x + h)\).

\[
g(x + h) = -2(x + h)^2 + (x + h) + 6
\]

First, expand \((x + h)^2\):

\[
(x + h)^2 = x^2 + 2xh + h^2
\]

Now, substitute this into the equation for \(g(x + h)\):

\[
g(x + h) = -2(x^2 + 2xh + h^2) + (x + h) + 6
\]

Distribute the \(-2\):

\[
g(x + h) = -2x^2 - 4xh - 2h^2 + x + h + 6
\]

So,

\[
g(x + h) = -2x^2 - 4xh - 2h^2 + x + h + 6
\]

### Step 2: Compute \(g(x + h) - g(x)\)
Now, subtract \(g(x)\) from \(g(x + h)\). Recall that \(g(x) = -2x^2 + x + 6\), so:

\[
g(x + h) - g(x) = \left(-2x^2 - 4xh - 2h^2 + x + h + 6\right) - \left(-2x^2 + x + 6\right)
\]

Simplify this expression:

\[
g(x + h) - g(x) = -2x^2 - 4xh - 2h^2 + x + h + 6 + 2x^2 - x - 6
\]

Now cancel out the like terms:

- The \(-2x^2\) and \(2x^2\) cancel.
- The \(x\) and \(-x\) cancel.
- The \(6\) and \(-6\) cancel.

We are left with:

\[
g(x + h) - g(x) = -4xh - 2h^2 + h
\]

### Step 3: Divide by \(h\)
Now, divide the entire expression by \(h\):

\[
\frac{g(x + h) - g(x)}{h} = \frac{-4xh - 2h^2 + h}{h}
\]

Simplify each term:

\[
\frac{g(x + h) - g(x)}{h} = \frac{-4xh}{h} + \frac{-2h^2}{h} + \frac{h}{h}
\]

This simplifies to:

\[
= -4x - 2h + 1
\]

### Final Answer:
\[
\frac{g(x+h) - g(x)}{h} = -4x - 2h + 1
\]

This is the simplified form of the difference quotient.

Would you like more details or clarifications on any step?

### Related Questions:
1. What is the geometric interpretation of the difference quotient?
2. How does the difference quotient relate to the derivative of \(g(x)\)?
3. What happens to \(\frac{g(x+h) - g(x)}{h}\) as \(h \to 0\)?
4. Can you find the derivative \(g'(x)\) from the result?
5. What is the significance of each term in the expression \(-4x - 2h + 1\)?

### Tip:
The difference quotient becomes the derivative when \(h \to 0\). This is a key concept in understanding rates of change and the slope of a function at a point.

Given g(x) = -2x^2 + x + 6, determine (g(x+h) - g(x))/h

Learn how to compute the difference quotient for the quadratic function g(x) = -2x^2 + x + 6. Follow detailed steps including calculating g(x+h), expanding the expression, and simplifying the result for (g(x+h) - g(x))/h.

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