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Let's solve the quadratic inequality \( 4(x + 2)^2 \leq 0 \).

### Step-by-step solution:

1. **Simplify the inequality:**

   The expression can be rewritten as:

   \[
   4(x + 2)^2 \leq 0
   \]

   Divide both sides by 4:

   \[
   (x + 2)^2 \leq 0
   \]

2. **Interpret the inequality:**

   - The square of a real number is always **non-negative**, meaning \( (x + 2)^2 \geq 0 \) for all \( x \in \mathbb{R} \).
   - The only way \( (x + 2)^2 = 0 \) is if \( x + 2 = 0 \), which gives \( x = -2 \).

3. **Conclusion:**

   The inequality \( (x + 2)^2 \leq 0 \) holds only when \( x = -2 \). For any other value of \( x \), \( (x + 2)^2 \) will be positive, and thus the inequality would not hold.

Thus, the solution set is \( \{ x \mid x = -2 \} \).

### Answer:

The correct answer is the first option: \( \{ x \mid x = -2 \} \).

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Would you like more details or have any questions? Here are 5 related questions to expand on this:

1. How would the solution change if the inequality were \( 4(x + 2)^2 < 0 \)?
2. What are the general steps to solve quadratic inequalities?
3. How would you solve \( 4(x + 2)^2 \geq 0 \)?
4. What is the difference between solving an inequality and an equation?
5. Can the inequality \( 4(x + 2)^2 \leq 0 \) have more than one solution?

**Tip**: Always remember that the square of any real number is non-negative. This fact is key in solving many quadratic inequalities.

What is the solution set of the quadratic inequality 4(x + 2)^2 ≤ 0?

Learn how to solve the quadratic inequality 4(x + 2)^2 ≤ 0 with a step-by-step solution. This problem explores the properties of quadratic expressions and inequalities, focusing on the non-negativity of squares. Suitable for students in Grades 9-12.

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