1. **State the problem:** Solve the exponential equation $$9^{x+1} = 243^{x+1}$$ for all values of $x$. 2. **Rewrite bases as powers of the same base:** $$9 = 3^2$$ $$243 = 3^5$$ 3. **Rewrite the equation using these powers:** $$\left(3^2\right)^{x+1} = \left(3^5\right)^{x+1}$$ 4. **Apply the power of a power rule:** $$3^{2(x+1)} = 3^{5(x+1)}$$ 5. **Since the bases are equal and nonzero, set exponents equal:** $$2(x+1) = 5(x+1)$$ 6. **Simplify the equation:** $$2x + 2 = 5x + 5$$ 7. **Bring all terms to one side:** $$2x + 2 - 5x - 5 = 0$$ $$-3x - 3 = 0$$ 8. **Solve for $x$:** $$-3x = 3$$ $$x = \frac{3}{-3}$$ $$x = -1$$ 9. **Check the solution:** Substitute $x = -1$ into the original equation: $$9^{(-1)+1} = 9^0 = 1$$ $$243^{(-1)+1} = 243^0 = 1$$ Both sides equal 1, so $x = -1$ is a valid solution. **Final answer:** $$x = -1$$