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The function provided is \( f(x) = -5x^2 - 3x + 8 \). We need to evaluate two expressions:

1. \( f(x + h) \)
2. \( \frac{f(x + h) - f(x)}{h} \)

Let's break this down.

### 1. Finding \( f(x + h) \)

To compute \( f(x + h) \), substitute \( x + h \) for \( x \) in the function:
\[
f(x + h) = -5(x + h)^2 - 3(x + h) + 8
\]
Now, expand this expression:

\[
(x + h)^2 = x^2 + 2xh + h^2
\]
So,
\[
f(x + h) = -5(x^2 + 2xh + h^2) - 3(x + h) + 8
\]
Expanding further:
\[
f(x + h) = -5x^2 - 10xh - 5h^2 - 3x - 3h + 8
\]

### 2. Finding \( \frac{f(x + h) - f(x)}{h} \)

Now, subtract \( f(x) \) from \( f(x + h) \):
\[
f(x + h) - f(x) = (-5x^2 - 10xh - 5h^2 - 3x - 3h + 8) - (-5x^2 - 3x + 8)
\]
Simplify this:
\[
f(x + h) - f(x) = -10xh - 5h^2 - 3h
\]
Finally, divide by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = \frac{-10xh - 5h^2 - 3h}{h}
\]
Simplify:
\[
\frac{f(x + h) - f(x)}{h} = -10x - 5h - 3
\]

### Final answers:
- \( f(x + h) = -5x^2 - 10xh - 5h^2 - 3x - 3h + 8 \)
- \( \frac{f(x + h) - f(x)}{h} = -10x - 5h - 3 \)

Would you like more details or clarifications on any of these steps?

Here are 5 related questions you might find useful:
1. How do you find the derivative of \( f(x) = -5x^2 - 3x + 8 \) using limits?
2. What happens as \( h \to 0 \) in the difference quotient \( \frac{f(x+h) - f(x)}{h} \)?
3. How do you find the tangent line to \( f(x) \) at a given point?
4. What is the significance of the difference quotient in calculus?
5. How does expanding \( (x + h)^2 \) help in simplifying the difference quotient?

**Tip:** The difference quotient \( \frac{f(x + h) - f(x)}{h} \) is the foundation for finding the derivative, which represents the slope of the tangent line to the function at any point \( x \).

For the function f(x) = -5x^2 - 3x + 8, evaluate and fully simplify f(x + h) and (f(x + h) - f(x)) / h.

Learn how to evaluate and fully simplify f(x + h) and the difference quotient for the quadratic function f(x) = -5x^2 - 3x + 8. This problem covers key algebraic concepts involved in calculating the difference quotient, a fundamental step in understanding derivatives.

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