The expression you provided seems to have a typo. Assuming you meant to write \(x^2 + 3x + 9\), let's analyze and solve it.

### Simplification and Solving

1. **Expression Given:**
   \[
   x^2 + 3x + 9
   \]

2. **Checking for Factorization:**
   The quadratic expression \(x^2 + 3x + 9\) cannot be factored easily as it doesn't have real roots (the discriminant is negative).

3. **Discriminant Calculation:**
   \[
   \Delta = b^2 - 4ac
   \]
   where \(a = 1\), \(b = 3\), and \(c = 9\):
   \[
   \Delta = 3^2 - 4 \cdot 1 \cdot 9 = 9 - 36 = -27
   \]
   Since \(\Delta < 0\), there are no real solutions. The roots are complex.

4. **Finding Complex Roots:**
   Using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{\Delta}}{2a}
   \]
   \[
   x = \frac{-3 \pm \sqrt{-27}}{2 \cdot 1}
   \]
   \[
   x = \frac{-3 \pm 3i\sqrt{3}}{2}
   \]
   So the solutions are:
   \[
   x = \frac{-3 + 3i\sqrt{3}}{2} \quad \text{and} \quad x = \frac{-3 - 3i\sqrt{3}}{2}
   \]

### Summary

The quadratic equation \(x^2 + 3x + 9 = 0\) has no real roots, but it has two complex roots:
\[
x = \frac{-3 + 3i\sqrt{3}}{2} \quad \text{and} \quad x = \frac{-3 - 3i\sqrt{3}}{2}
\]

Would you like more details or have any questions about this solution?

### Further Exploration Questions
1. What is the significance of the discriminant in a quadratic equation?
2. How do you find the roots of a quadratic equation with real coefficients?
3. Can a quadratic equation have one real and one complex root?
4. What are some real-world applications of solving quadratic equations?
5. How does the quadratic formula derive from completing the square?
6. What is the geometric representation of complex roots?
7. How do you solve a quadratic equation by factoring when it is factorable?
8. What are some methods to simplify complex numbers in algebraic form?

### Tip
When solving quadratic equations, always check the discriminant first. It quickly tells you the nature of the roots (real and distinct, real and repeated, or complex).

Learn how to solve the quadratic equation x^2 + 3x + 9 with complex roots. Understand the role of the discriminant in determining the nature of roots in quadratic equations. Suitable for students in Grades 10-12.

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