1. **State the problem:** Simplify or analyze the quadratic expression $x^2 + 5x + 10$. 2. **Recall the quadratic formula:** To find roots of $ax^2 + bx + c = 0$, use $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=5$, and $c=10$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 1 \times 10 = 25 - 40 = -15$$ 4. **Interpret the discriminant:** Since $\Delta < 0$, the quadratic has no real roots, only complex roots. 5. **Find the roots:** $$x = \frac{-5 \pm \sqrt{-15}}{2} = \frac{-5 \pm i\sqrt{15}}{2}$$ 6. **Summary:** The quadratic $x^2 + 5x + 10$ cannot be factored over the real numbers and has complex roots $\frac{-5 \pm i\sqrt{15}}{2}$.