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Let's solve the given problem step by step:

We are working with the quadratic function:

\[
f(x) = -x^2 - 8x
\]

### (e) Determine where the quadratic function is increasing and decreasing.
To solve this, we need to find the critical points and analyze the intervals of increasing and decreasing behavior.

1. **First derivative**:
   To find where the function is increasing or decreasing, we first find the derivative of \( f(x) \).

   \[
   f'(x) = \frac{d}{dx}(-x^2 - 8x) = -2x - 8
   \]

2. **Critical points**:
   Set \( f'(x) = 0 \) to find critical points.

   \[
   -2x - 8 = 0
   \]
   \[
   -2x = 8
   \]
   \[
   x = -4
   \]

   So, the critical point is \( x = -4 \).

3. **Test intervals**:
   Now, we'll analyze the sign of \( f'(x) \) on the intervals \( (-\infty, -4) \) and \( (-4, \infty) \):

   - For \( x < -4 \), let's pick \( x = -5 \):
     \[
     f'(-5) = -2(-5) - 8 = 10 - 8 = 2 \quad (\text{positive, so } f(x) \text{ is increasing})
     \]

   - For \( x > -4 \), let's pick \( x = 0 \):
     \[
     f'(0) = -2(0) - 8 = -8 \quad (\text{negative, so } f(x) \text{ is decreasing})
     \]

   Therefore, \( f(x) \) is increasing on \( (-\infty, -4] \) and decreasing on \( [-4, \infty) \).

   **Answer**:
   - Increasing on \( (-\infty, -4] \)
   - Decreasing on \( [-4, \infty) \)

### (f) Determine where \( f(x) > 0 \) and where \( f(x) < 0 \).

We need to find when the quadratic function is positive and when it is negative.

1. **Find the roots**:
   Set \( f(x) = 0 \) to find the x-intercepts.

   \[
   -x^2 - 8x = 0
   \]
   Factor the equation:
   \[
   -x(x + 8) = 0
   \]
   So, the roots are \( x = 0 \) and \( x = -8 \).

2. **Sign analysis**:
   The quadratic function opens downward (since the coefficient of \( x^2 \) is negative), so it is positive between the two roots and negative outside this interval.

   - \( f(x) > 0 \) on \( (-8, 0) \)
   - \( f(x) < 0 \) on \( (-\infty, -8) \cup (0, \infty) \)

**Answer**:
Select option **B**:  
\( f(x) > 0 \) on \( (-8, 0) \) and \( f(x) < 0 \) on \( (-\infty, -8) \cup (0, \infty) \).

---

Would you like more details on any step or explanation?

Here are 5 related questions:
1. How do you find the vertex of a quadratic function?
2. Why is the function positive between the roots of a downward-opening parabola?
3. What is the role of the discriminant in determining the number of roots?
4. How do you solve inequalities involving quadratic functions?
5. What are the key properties of a quadratic function?

**Tip**: For any quadratic function, the sign of the leading coefficient determines whether the parabola opens upward or downward.

For the quadratic function f(x) = -x^2 - 8x, determine where the quadratic function is increasing and decreasing. Also, determine where f(x) > 0 and where f(x) < 0.

Learn how to find the increasing and decreasing intervals of the quadratic function f(x) = -x^2 - 8x using the first derivative test and critical points. Also, discover where the function is positive and negative through sign analysis and factorization. This solution is suitable for students in Grades 9-12.

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