The inequality you're working with is:

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

First, factor the quadratic expression on the left-hand side:

\[
x^2 + 4x + 4 = (x + 2)^2
\]

So, the inequality becomes:

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

### Step 1: Analyze the square
Since \((x + 2)^2\) is a square, it is always non-negative, meaning \((x + 2)^2 \geq 0\) for all \(x\). The only value for which \((x + 2)^2 = 0\) is when \(x + 2 = 0\), i.e., \(x = -2\).

Thus, \((x + 2)^2 \leq 0\) only holds when \(x = -2\), because for any other value of \(x\), \((x + 2)^2 > 0\).

### Step 2: Conclusion
The solution to the inequality \((x + 2)^2 \leq 0\) is:

\[
x = -2
\]

### Final Answer:
The solution is \(x = -2\).

Do you want any further details or have any questions?

Here are 5 related questions:
1. How do you solve quadratic inequalities in general?
2. What happens if the inequality was strict \((x^2 + 4x + 4 < 0)\)?
3. How do you solve inequalities with absolute values?
4. Can quadratic inequalities have no real solution?
5. How can you graph the solution of quadratic inequalities?

**Tip:** When solving quadratic inequalities, check whether the expression can be factored to make analysis easier.

Discover how to solve the quadratic inequality x^2 + 4x + 4 ≤ 0 using factoring and key algebraic concepts. This problem focuses on analyzing squares and solving quadratic inequalities. Ideal for students in Grades 9-11.

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