1. **State the problem:** Solve the quadratic equation $$3x^2 - 2x - 5 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equation is in the form $$ax^2 + bx + c = 0$$. 3. **Identify coefficients:** Here, $$a=3$$, $$b=-2$$, and $$c=-5$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 3 \times (-5) = 4 + 60 = 64$$. 5. **Apply the quadratic formula:** $$x = \frac{-(-2) \pm \sqrt{64}}{2 \times 3} = \frac{2 \pm 8}{6}$$. 6. **Find the two solutions:** - For the plus sign: $$x = \frac{2 + 8}{6} = \frac{10}{6} = \frac{\cancel{10}}{\cancel{6}} = \frac{5}{3}$$. - For the minus sign: $$x = \frac{2 - 8}{6} = \frac{-6}{6} = \frac{\cancel{-6}}{\cancel{6}} = -1$$. 7. **Final answer:** The solutions are $$x = \frac{5}{3}$$ and $$x = -1$$.