1. **State the problem:** Solve the quadratic equation $$3x^2 + 2x - 21 = 0$$. 2. **Formula used:** The quadratic formula for solving $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 3$$, $$b = 2$$, and $$c = -21$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4 \times 3 \times (-21) = 4 + 252 = 256$$. 5. **Find the square root of the discriminant:** $$\sqrt{256} = 16$$. 6. **Apply the quadratic formula:** $$x = \frac{-2 \pm 16}{2 \times 3} = \frac{-2 \pm 16}{6}$$. 7. **Calculate the two solutions:** - $$x_1 = \frac{-2 + 16}{6} = \frac{14}{6} = \frac{7}{3}$$ - $$x_2 = \frac{-2 - 16}{6} = \frac{-18}{6} = -3$$. **Final answer:** $$x = \frac{7}{3}$$ or $$x = -3$$.