1. **State the problem:** Solve the quadratic equation $3x^2 - 7x = 20$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$3x^2 - 7x - 20 = 0$$ 3. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Identify coefficients:** Here, $a=3$, $b=-7$, and $c=-20$. 5. **Calculate the discriminant:** $$b^2 - 4ac = (-7)^2 - 4 \times 3 \times (-20) = 49 + 240 = 289$$ 6. **Calculate the square root of the discriminant:** $$\sqrt{289} = 17$$ 7. **Apply the quadratic formula:** $$x = \frac{-(-7) \pm 17}{2 \times 3} = \frac{7 \pm 17}{6}$$ 8. **Find the two solutions:** - For the plus sign: $$x = \frac{7 + 17}{6} = \frac{24}{6} = 4$$ - For the minus sign: $$x = \frac{7 - 17}{6} = \frac{-10}{6} = \frac{-5}{3}$$ 9. **Final answer:** The solutions are $$\boxed{\left\{ -\frac{5}{3}, 4 \right\}}$$ This corresponds to option D.