https://file.aiquickdraw.com/mathbot/file/1727073162828nk1xi6wr.jpg

Let's solve the problems presented in the image one by one.

---

### 1. Solve the equation:
\[
\sqrt[3]{5x + x^2 - 8} = -2
\]

**Solution:**

- Start by cubing both sides to eliminate the cube root:
  \[
  5x + x^2 - 8 = (-2)^3
  \]
  Simplifying the right side:
  \[
  5x + x^2 - 8 = -8
  \]
  
- Add 8 to both sides to cancel the constant:
  \[
  5x + x^2 = 0
  \]
  
- Factor the equation:
  \[
  x(5 + x) = 0
  \]

- This gives two solutions:
  \[
  x = 0 \quad \text{or} \quad x = -5
  \]

**Solution set:**
\[
x = 0 \quad \text{or} \quad x = -5
\]

---

### 2. Find the domain and zeros of the function:
\[
y = \frac{\sqrt{|x| - 1}}{x^2 - 3}
\]

**Solution:**

- **Domain:**
  - The expression inside the square root must be non-negative:
    \[
    |x| - 1 \geq 0 \quad \Rightarrow \quad |x| \geq 1
    \]
    This gives two intervals:
    \[
    x \leq -1 \quad \text{or} \quad x \geq 1
    \]
  - The denominator cannot be zero:
    \[
    x^2 - 3 \neq 0 \quad \Rightarrow \quad x \neq \pm\sqrt{3}
    \]
  - Therefore, the domain of the function is:
    \[
    (-\infty, -\sqrt{3}) \cup (-\sqrt{3}, -1] \cup [1, \sqrt{3}) \cup (\sqrt{3}, \infty)
    \]

- **Zeros:**
  - For the function to be zero, the numerator must be zero:
    \[
    \sqrt{|x| - 1} = 0 \quad \Rightarrow \quad |x| - 1 = 0 \quad \Rightarrow \quad |x| = 1
    \]
    Thus, \( x = \pm 1 \).

**Zeros:**
\[
x = -1 \quad \text{and} \quad x = 1
\]

---

### 3. Solve the inequality using the interval method:
\[
\frac{x^7(x^2 + 4x + 4)}{x^2 - 3} \leq 0
\]

**Solution:**

- Factor the quadratic in the numerator:
  \[
  x^2 + 4x + 4 = (x + 2)^2
  \]
  So the inequality becomes:
  \[
  \frac{x^7(x + 2)^2}{x^2 - 3} \leq 0
  \]

- Critical points occur where the expression equals zero or is undefined:
  - \( x^7 = 0 \Rightarrow x = 0 \)
  - \( (x + 2)^2 = 0 \Rightarrow x = -2 \)
  - \( x^2 - 3 = 0 \Rightarrow x = \pm\sqrt{3} \)

- These critical points divide the real line into intervals. We need to test each interval to see where the expression is negative or zero:
  \[
  (-\infty, -\sqrt{3}), (-\sqrt{3}, -2), (-2, 0), (0, \sqrt{3}), (\sqrt{3}, \infty)
  \]
  - Analyze the sign of the expression in each interval and consider the zeros.

- **Final solution:**
  The inequality is satisfied in the following intervals:
  \[
  (-\infty, -\sqrt{3}) \cup (-2, 0] \cup (0, \sqrt{3})
  \]

---

Would you like more details on any of these solutions? 

Here are 5 related questions for practice:

1. How would the equation change if the cube root in problem 1 were replaced with a square root?
2. What is the range of the function in problem 2?
3. How would the solution change in problem 3 if the inequality were strict \( < 0 \)?
4. Can you solve \( \sqrt[3]{5x + x^2 - 8} = 3 \)?
5. How would you find the maximum or minimum of the function in problem 2?

**Tip:** Always check the domain of a function before solving, especially if there are square roots or denominators.

Solve the following: 
1. \( \sqrt[3]{5x + x^2 - 8} = -2 \)
2. Find the domain and zeros of the function \( y = \frac{\sqrt{|x| - 1}}{x^2 - 3} \)
3. Solve the inequality \( \frac{x^7(x^2 + 4x + 4)}{x^2 - 3} \leq 0 \)

Solve a cubic equation, determine the domain and zeros of a complex function, and solve an inequality using the interval method. This detailed solution covers cubic roots, quadratic expressions, and function domain restrictions, suitable for students in Grades 10-12.

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