1. **State the problem:** Solve the quadratic equation $$10 - 9x^2 + 4x = -6x^2$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$10 - 9x^2 + 4x + 6x^2 = 0$$ 3. **Combine like terms:** $$10 - 3x^2 + 4x = 0$$ 4. **Rewrite in standard quadratic form:** $$-3x^2 + 4x + 10 = 0$$ 5. **Multiply both sides by $-1$ to make the leading coefficient positive:** $$\cancel{-3}x^2 + \cancel{4}x + \cancel{10} = 0 \implies 3x^2 - 4x - 10 = 0$$ 6. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=-4$, and $c=-10$. 7. **Calculate the discriminant:** $$b^2 - 4ac = (-4)^2 - 4(3)(-10) = 16 + 120 = 136$$ 8. **Substitute values into the quadratic formula:** $$x = \frac{-(-4) \pm \sqrt{136}}{2 \times 3} = \frac{4 \pm \sqrt{136}}{6}$$ 9. **Simplify $\sqrt{136}$:** $$\sqrt{136} = \sqrt{4 \times 34} = 2\sqrt{34}$$ 10. **Final simplified solution:** $$x = \frac{4 \pm 2\sqrt{34}}{6} = \frac{2(2 \pm \sqrt{34})}{6} = \frac{2 \pm \sqrt{34}}{3}$$ 11. **Check answer choices:** None exactly match this form, but choice D is: $$x = \frac{-2 \pm \sqrt{34}}{-3}$$ Multiply numerator and denominator of choice D by $-1$: $$x = \frac{2 \mp \sqrt{34}}{3}$$ which matches our solution (just the $\pm$ sign order reversed, which is equivalent). **Therefore, the correct answer is D.**