1. **State the problem:** Solve the inequality $$2x^2 - 3x - 16 > 5 - 2x$$ for $x$. 2. **Rewrite the inequality:** Move all terms to one side to set the inequality to zero: $$2x^2 - 3x - 16 - 5 + 2x > 0$$ 3. **Simplify the expression:** $$2x^2 - 3x + 2x - 16 - 5 > 0$$ $$2x^2 - x - 21 > 0$$ 4. **Find the roots of the quadratic equation:** Solve $$2x^2 - x - 21 = 0$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-1$, and $c=-21$. 5. **Calculate the discriminant:** $$\Delta = (-1)^2 - 4 \times 2 \times (-21) = 1 + 168 = 169$$ 6. **Calculate the roots:** $$x = \frac{-(-1) \pm \sqrt{169}}{2 \times 2} = \frac{1 \pm 13}{4}$$ 7. **Find the two roots:** $$x_1 = \frac{1 - 13}{4} = \frac{-12}{4} = -3$$ $$x_2 = \frac{1 + 13}{4} = \frac{14}{4} = 3.5$$ 8. **Determine the intervals where the inequality holds:** Since the parabola opens upwards ($a=2 > 0$), the quadratic is greater than zero outside the roots: $$x < -3 \quad \text{or} \quad x > 3.5$$ **Final answer:** $$\boxed{x < -3 \text{ or } x > 3.5}$$