1. **State the problem:** Solve the quadratic equation $2x^2 + x - e^2 = 0$ for $x$. 2. **Recall the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=1$, and $c=-e^2$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 1^2 - 4 \times 2 \times (-e^2) = 1 + 8e^2$$ 4. **Apply the quadratic formula:** $$x = \frac{-1 \pm \sqrt{1 + 8e^2}}{2 \times 2} = \frac{-1 \pm \sqrt{1 + 8e^2}}{4}$$ 5. **Final answer:** $$x = \frac{-1 + \sqrt{1 + 8e^2}}{4} \quad \text{or} \quad x = \frac{-1 - \sqrt{1 + 8e^2}}{4}$$ These are the two solutions to the quadratic equation.