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To solve the given problem step-by-step, let's analyze the function \( f(x) = |(x + 2)^3| - 1 \) and answer each of the questions provided.

1. **Finding the zeros of the function \( f(x) \):**

   The function \( f(x) = |(x + 2)^3| - 1 \) will be zero when:
   \[
   |(x + 2)^3| - 1 = 0
   \]
   This simplifies to:
   \[
   |(x + 2)^3| = 1
   \]
   Solving for \( x \), we consider two cases:

   - **Case 1**: \( (x + 2)^3 = 1 \)
     - Taking the cube root on both sides, \( x + 2 = 1 \Rightarrow x = -1 \).
   - **Case 2**: \( (x + 2)^3 = -1 \)
     - Taking the cube root, \( x + 2 = -1 \Rightarrow x = -3 \).

   Therefore, the zeros of \( f(x) \) are at \( x = -1 \) and \( x = -3 \).

2. **Finding the y-intercept of the function:**

   The y-intercept occurs when \( x = 0 \):
   \[
   f(0) = |(0 + 2)^3| - 1 = |8| - 1 = 8 - 1 = 7
   \]
   So, the y-intercept is at \( (0, 7) \).

3. **Recording the extrema (maximums) of the function in the interval \([-4, 1]\):**

   To find the extrema, we should examine the behavior of \( f(x) \) within \([-4, 1]\) and determine if there are any critical points within this interval.

   - At \( x = -4 \):
     \[
     f(-4) = |(-4 + 2)^3| - 1 = |(-2)^3| - 1 = | -8 | - 1 = 8 - 1 = 7
     \]
   - At \( x = -3 \) (a zero, so this is a critical point):
     \[
     f(-3) = 0
     \]
   - At \( x = -1 \) (another zero):
     \[
     f(-1) = 0
     \]
   - At \( x = 1 \):
     \[
     f(1) = |(1 + 2)^3| - 1 = |3^3| - 1 = |27| - 1 = 26
     \]

   Therefore, the maximum value in \([-4, 1]\) occurs at \( x = 1 \), where \( f(1) = 26 \).

4. **Recording the interval where the function is decreasing and positive in the interval \([-4, 1]\):**

   To determine where the function is decreasing, we examine the derivative of \( f(x) \) and the behavior of \( |(x + 2)^3| \). The function \( f(x) \) decreases in the interval \([-4, -3]\), as \( (x+2)^3 \) is negative and becomes less negative as \( x \) approaches -3.

5. **Identifying where the function is positive only for \( x \leq 0 \):**

   From the zeros calculated in part 1, the function \( f(x) \) is positive in the interval \([-4, -3) \cup (-3, -1) \cup (-1, 1]\) where it is strictly above zero.

6. **Identifying where the function is positive only for \( x > 0 \):**

   In the interval \( x > 0 \), the function is positive for \( x > 0 \) as \( f(x) \to \infty \) as \( x \) grows.

Find the zeros of the function, the y-intercept, the extrema in the interval [-4, 1], the decreasing intervals, and the positivity conditions for the function f(x) = |(x + 2)^3| - 1.

Step-by-step solution for analyzing the function f(x) = |(x + 2)^3| - 1, covering zeros, y-intercept, extrema in [-4, 1], decreasing intervals, and conditions for positivity. Suitable for Grade 10 students studying algebra and function properties.

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