1. **State the problem:** Solve the quadratic inequality $$2x^2 - 3x \leq 10$$ and analyze the graph of the quadratic function $$y = 2x^2 - 3x$$. 2. **Rewrite the inequality:** Move all terms to one side to set the inequality to zero: $$2x^2 - 3x - 10 \leq 0$$ 3. **Find the roots of the quadratic equation:** Solve $$2x^2 - 3x - 10 = 0$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=2$$, $$b=-3$$, and $$c=-10$$. 4. **Calculate the discriminant:** $$\Delta = (-3)^2 - 4 \times 2 \times (-10) = 9 + 80 = 89$$ 5. **Find the roots:** $$x = \frac{3 \pm \sqrt{89}}{4}$$ 6. **Determine the solution set:** Since the parabola opens upwards (because $$a=2 > 0$$), the quadratic expression is less than or equal to zero between the roots: $$\frac{3 - \sqrt{89}}{4} \leq x \leq \frac{3 + \sqrt{89}}{4}$$ 7. **Graph description:** The graph of $$y = 2x^2 - 3x$$ is a parabola opening upwards. It intersects the x-axis at the roots found above. The region where the parabola lies below or on the x-axis corresponds to the solution of the inequality. **Final answer:** $$\boxed{\frac{3 - \sqrt{89}}{4} \leq x \leq \frac{3 + \sqrt{89}}{4}}$$