1. **State the problem:** Simplify or analyze the quadratic expression $x^2 + x + 1$. 2. **Recall the quadratic formula:** For any quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=1$, and $c=1$ in this case. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 1^2 - 4 \times 1 \times 1 = 1 - 4 = -3$$ 4. **Interpret the discriminant:** Since $\Delta < 0$, the quadratic has no real roots; it has two complex conjugate roots. 5. **Find the roots:** $$x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}$$ 6. **Summary:** The quadratic $x^2 + x + 1$ cannot be factored over the real numbers and has complex roots $\frac{-1 + i\sqrt{3}}{2}$ and $\frac{-1 - i\sqrt{3}}{2}$. This expression is always positive for real $x$ because its graph is a parabola opening upwards with no real zeros.