1. **State the problem:** Solve the quadratic equation $$3x^2 - 7x - 268 = 0$$. 2. **Formula used:** To solve a quadratic equation $$ax^2 + bx + c = 0$$, use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=-7$, and $c=-268$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-7)^2 - 4 \times 3 \times (-268) = 49 + 3216 = 3265$$ 4. **Apply the quadratic formula:** $$x = \frac{-(-7) \pm \sqrt{3265}}{2 \times 3} = \frac{7 \pm \sqrt{3265}}{6}$$ 5. **Simplify the square root if possible:** $3265 = 5 \times 653$, and 653 is prime, so no simplification. 6. **Final solutions:** $$x_1 = \frac{7 + \sqrt{3265}}{6}$$ $$x_2 = \frac{7 - \sqrt{3265}}{6}$$ These are the exact solutions to the quadratic equation.