1. **State the problem:** Solve the quadratic equation $-x^2 + 3x + 10 = 0$ for $x$. 2. **Rewrite the equation:** Multiply both sides by $-1$ to get a standard form: $$x^2 - 3x - 10 = 0$$ 3. **Identify coefficients:** Here, $a = 1$, $b = -3$, and $c = -10$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-3)^2 - 4(1)(-10) = 9 + 40 = 49$$ 6. **Find the roots:** $$x = \frac{-(-3) \pm \sqrt{49}}{2(1)} = \frac{3 \pm 7}{2}$$ 7. **Calculate each solution:** - For $+$ sign: $x = \frac{3 + 7}{2} = \frac{10}{2} = 5$ - For $-$ sign: $x = \frac{3 - 7}{2} = \frac{-4}{2} = -2$ 8. **Final answer:** The solutions are $x = 5$ and $x = -2$.