1. **State the problem:** Solve the inequality $$3^8 \leq -2x^2$$ for $x$. 2. **Analyze the inequality:** The left side is $3^8$, a positive constant, and the right side is $-2x^2$, which is always non-positive (zero or negative) because $x^2 \geq 0$ and multiplied by $-2$ makes it non-positive. 3. **Evaluate $3^8$:** $$3^8 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 6561$$ 4. **Rewrite the inequality:** $$6561 \leq -2x^2$$ 5. **Consider the right side:** Since $-2x^2 \leq 0$ for all real $x$, the right side is at most zero. 6. **Compare both sides:** The left side is $6561 > 0$, the right side is $\leq 0$. So the inequality $$6561 \leq -2x^2$$ cannot be true for any real $x$ because a positive number cannot be less than or equal to a non-positive number. 7. **Conclusion:** There is no real solution to the inequality. **Final answer:** $$\boxed{\text{No real solutions}}$$