1. **State the problem:** Factor the quadratic expression $x^2 + 2x + 3$. 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, $a=1$, $b=2$, and $c=3$. We need two numbers that multiply to $1 \times 3 = 3$ and add to $2$. 4. **Check possible pairs:** The pairs of factors of 3 are (1, 3) and (-1, -3). Neither pair sums to 2. 5. **Conclusion:** Since no real factors satisfy the conditions, the quadratic does not factor over the real numbers. 6. **Alternative:** Use the quadratic formula to find roots: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{4 - 12}}{2} = \frac{-2 \pm \sqrt{-8}}{2} = -1 \pm i\sqrt{2}$$ 7. **Final factorization over complex numbers:** $$x^2 + 2x + 3 = (x - (-1 + i\sqrt{2}))(x - (-1 - i\sqrt{2})) = (x + 1 - i\sqrt{2})(x + 1 + i\sqrt{2})$$