1. **State the problem:** Solve the quadratic equation $$-3x^2 + 8x - 7 = 0$$ using the quadratic formula. 2. **Quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = -3$$, $$b = 8$$, and $$c = -7$$. 4. **Fill in the quadratic formula:** $$x = \frac{-8 \pm \sqrt{8^2 - 4(-3)(-7)}}{2(-3)}$$ 5. **Simplify inside the square root:** $$8^2 = 64$$ $$4 \times (-3) \times (-7) = 84$$ 6. **Calculate the discriminant:** $$64 - 84 = -20$$ 7. **Rewrite the formula with the discriminant:** $$x = \frac{-8 \pm \sqrt{-20}}{-6}$$ 8. **Simplify the square root of a negative number:** $$\sqrt{-20} = \sqrt{-1 \times 20} = i\sqrt{20} = i \times 2\sqrt{5}$$ 9. **Substitute back:** $$x = \frac{-8 \pm 2i\sqrt{5}}{-6}$$ 10. **Simplify the fraction by dividing numerator and denominator by 2:** $$x = \frac{\cancel{-8} \pm \cancel{2}i\sqrt{5}}{\cancel{-6}} = \frac{-4 \pm i\sqrt{5}}{-3}$$ 11. **Simplify signs:** Dividing numerator and denominator by -1: $$x = \frac{4 \mp i\sqrt{5}}{3}$$ 12. **Final solution set:** $$x = \left\{ \frac{4 \pm i\sqrt{5}}{3} \right\}$$ 13. **Match with options:** Option A matches the solution. **Answer:** (a) $$x = \frac{-8 \pm \sqrt{8^2 - 4(-3)(-7)}}{2(-3)}$$ (b) The correct solution is option A: $$x = \left\{ \frac{4 \pm i\sqrt{5}}{3} \right\}$$