1. **State the problem:** Solve the equation $$9^x = 3^{2x+1}$$ for $x$. 2. **Rewrite the bases:** Note that $9$ can be written as $3^2$, so rewrite the equation as $$\left(3^2\right)^x = 3^{2x+1}$$. 3. **Apply the power of a power rule:** $$\left(3^2\right)^x = 3^{2x}$$, so the equation becomes $$3^{2x} = 3^{2x+1}$$. 4. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal: $$2x = 2x + 1$$. 5. **Solve for $x$:** Subtract $2x$ from both sides: $$0 = 1$$, which is a contradiction. 6. **Conclusion:** There is no solution to the equation because the exponents cannot be equal. **Final answer:** No solution.