1. **State the problem:** Solve the inequality $2x^2 + 7x - 4 > 0$. 2. **Formula and rules:** To solve a quadratic inequality, first find the roots of the quadratic equation $2x^2 + 7x - 4 = 0$. 3. **Find the roots using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=7$, and $c=-4$. 4. Calculate the discriminant: $$\Delta = b^2 - 4ac = 7^2 - 4 \times 2 \times (-4) = 49 + 32 = 81$$ 5. Calculate the roots: $$x = \frac{-7 \pm \sqrt{81}}{2 \times 2} = \frac{-7 \pm 9}{4}$$ 6. Find each root: $$x_1 = \frac{-7 + 9}{4} = \frac{2}{4} = \frac{\cancel{2}}{\cancel{4}} = \frac{1}{2}$$ $$x_2 = \frac{-7 - 9}{4} = \frac{-16}{4} = -4$$ 7. **Analyze the inequality:** Since $a=2 > 0$, the parabola opens upwards. 8. The quadratic expression is positive outside the roots and negative between them. 9. Therefore, the solution to $2x^2 + 7x - 4 > 0$ is: $$x < -4 \quad \text{or} \quad x > \frac{1}{2}$$ **Final answer:** $$\boxed{x < -4 \text{ or } x > \frac{1}{2}}$$