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Let's work through the given function \( f(x) = 4(x + 4)^2 - 5 \) to find the intervals of increase and decrease.

### Step 1: Find the derivative
To determine where the function is increasing or decreasing, we first find its derivative.

\[
f(x) = 4(x + 4)^2 - 5
\]

Using the chain rule to differentiate:

\[
f'(x) = 4 \cdot 2(x + 4) \cdot 1 = 8(x + 4)
\]

So, the derivative is:

\[
f'(x) = 8(x + 4)
\]

### Step 2: Find the critical points
Set the derivative equal to zero to find critical points:

\[
8(x + 4) = 0
\]

Solving for \( x \):

\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

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

### Step 3: Determine the sign of \( f'(x) \)
To find where the function is increasing or decreasing, we analyze the sign of \( f'(x) \) on either side of \( x = -4 \).

- For \( x < -4 \), say \( x = -5 \):
  \[
  f'(-5) = 8(-5 + 4) = 8(-1) = -8
  \]
  So, \( f'(x) < 0 \) for \( x < -4 \), meaning the function is **decreasing** in this interval.

- For \( x > -4 \), say \( x = -3 \):
  \[
  f'(-3) = 8(-3 + 4) = 8(1) = 8
  \]
  So, \( f'(x) > 0 \) for \( x > -4 \), meaning the function is **increasing** in this interval.

### Step 4: Conclusion
- The function is **decreasing** on \( (-\infty, -4) \).
- The function is **increasing** on \( (-4, \infty) \).

### Final Answer:
- **Interval of increase**: \( (-4, \infty) \)
- **Interval of decrease**: \( (-\infty, -4) \)

Let me know if you have any further questions, or if you'd like more details!

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Here are 5 related questions for further exploration:
1. How would the intervals change if the function were \( f(x) = 4(x + 4)^2 + 5 \)?
2. What is the significance of the critical point \( x = -4 \) in terms of the graph's shape?
3. How do we determine if a critical point is a local maximum or minimum?
4. How would we solve for intervals of concavity instead of increasing/decreasing?
5. What happens to the derivative of a function when there is a change in the coefficient of \( x^2 \)?

**Tip**: When analyzing the behavior of a function, always look for critical points and test the sign of the derivative around them to determine increasing and decreasing intervals.

Determine the interval(s) on which the function f(x) = 4(x + 4)^2 - 5 is increasing and decreasing.

Determine the intervals of increase and decrease for the function f(x) = 4(x + 4)^2 - 5 using calculus. This problem involves finding the derivative and applying the first derivative test, suitable for Calculus 1 students.

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