1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 9$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Apply the formula:** Here, $a=1$, $b=5$, and $c=9$. We need two numbers that multiply to $1 \times 9 = 9$ and add to $5$. 4. **Check possible pairs:** The pairs of factors of 9 are (1,9) and (3,3). Neither pair sums to 5. 5. **Conclusion:** Since no integer factors satisfy the conditions, the quadratic does not factor nicely over the integers. 6. **Alternative approach - quadratic formula:** Use $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{25 - 36}}{2} = \frac{-5 \pm \sqrt{-11}}{2}$$ 7. **Final answer:** The quadratic has complex roots and cannot be factored over the real numbers. Its factorization over complex numbers is $$\left(x - \frac{-5 + \sqrt{-11}}{2}\right)\left(x - \frac{-5 - \sqrt{-11}}{2}\right)$$.