1. **State the problem:** Solve the quadratic inequality $$-x^2 + 2x - 10 \geq 0$$. 2. **Rewrite the inequality:** It is often easier to analyze when the quadratic term is positive, so multiply both sides by $$-1$$, remembering to reverse the inequality sign: $$-x^2 + 2x - 10 \geq 0 \implies x^2 - 2x + 10 \leq 0$$. 3. **Analyze the quadratic:** The quadratic is $$x^2 - 2x + 10$$. To understand where it is less than or equal to zero, find its discriminant: $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 1 \times 10 = 4 - 40 = -36$$. 4. **Interpret the discriminant:** Since $$\Delta < 0$$, the quadratic has no real roots and does not cross the x-axis. 5. **Determine the sign of the quadratic:** Because the leading coefficient $$a=1 > 0$$, the parabola opens upwards and is always positive. 6. **Conclusion:** Since $$x^2 - 2x + 10 > 0$$ for all real $$x$$, the inequality $$x^2 - 2x + 10 \leq 0$$ has no solution. 7. **Return to original inequality:** Therefore, the original inequality $$-x^2 + 2x - 10 \geq 0$$ also has no solution. **Final answer:** No real values of $$x$$ satisfy the inequality $$-x^2 + 2x - 10 \geq 0$$.