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.