Let's solve the inequality:

\[ x^2 + 6x + 9 > 0 \]

### Step 1: Factor the quadratic expression
The expression \( x^2 + 6x + 9 \) can be factored as:

\[
x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2
\]

So, the inequality becomes:

\[
(x + 3)^2 > 0
\]

### Step 2: Analyze the inequality
The expression \( (x + 3)^2 \) represents a perfect square. A perfect square is always non-negative (i.e., greater than or equal to 0) for all real numbers.

The only time \( (x + 3)^2 = 0 \) is when \( x = -3 \).

So, for the inequality \( (x + 3)^2 > 0 \) to be true, \( x + 3 \) must not equal 0, which means \( x \neq -3 \).

### Step 3: Conclusion
The inequality holds for all real values of \( x \) except \( x = -3 \).

**Solution:**

\[
x \in (-\infty, -3) \cup (-3, \infty)
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the solution to the inequality \(x^2 + 4x + 4 > 0\)?
2. How do you solve a quadratic inequality when it cannot be factored easily?
3. What is the graphical interpretation of \(x^2 + 6x + 9 > 0\)?
4. How does the nature of the discriminant affect the solution to quadratic inequalities?
5. Can a quadratic inequality ever have no real solutions?

**Tip:** When solving quadratic inequalities, always factor the expression if possible, then consider where the expression equals zero and its sign on intervals around the zeros.

Learn how to solve the quadratic inequality x^2 + 6x + 9 > 0 with a step-by-step guide. This solution involves factoring the quadratic expression into (x + 3)^2 and analyzing the inequality. Suitable for students in Grades 9-11.

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