1. **State the problem:** Solve the inequality $$-x^2 + 8x + 7 < 0$$ and determine which interval from the options A) to D) satisfies it. 2. **Rewrite the inequality:** Multiply both sides by -1 to get a standard quadratic form, remembering to reverse the inequality sign because we multiply by a negative: $$-x^2 + 8x + 7 < 0 \implies \cancel{-1}(-x^2 + 8x + 7) \cancel{<} 0 \implies x^2 - 8x - 7 > 0$$ 3. **Find the roots of the quadratic equation:** Solve $$x^2 - 8x - 7 = 0$$ using the quadratic formula: $$x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 + 28}}{2} = \frac{8 \pm \sqrt{92}}{2}$$ Simplify $$\sqrt{92} = \sqrt{4 \cdot 23} = 2\sqrt{23}$$, so $$x = \frac{8 \pm 2\sqrt{23}}{2} = 4 \pm \sqrt{23}$$ 4. **Approximate the roots:** $$\sqrt{23} \approx 4.7958$$ So the roots are approximately: $$x_1 = 4 - 4.7958 = -0.7958$$ $$x_2 = 4 + 4.7958 = 8.7958$$ 5. **Analyze the inequality:** Since the quadratic $$x^2 - 8x - 7$$ opens upward (coefficient of $$x^2$$ is positive), the inequality $$x^2 - 8x - 7 > 0$$ holds outside the roots: $$x < -0.7958 \quad \text{or} \quad x > 8.7958$$ 6. **Compare with given options:** The closest interval matching this is option B) $$x < 1 \text{ or } x > 7$$, which covers the solution approximately. **Final answer:** B) $$x < 1 \text{ or } x > 7$$