What value of x satisfies 9^x+2 = 27^x-1?

Rewrite both sides with base 3. Since $9 = 3^2$ and $27 = 3^3$, we get $3^{2(x+2)} = 3^{3(x-1)}$. Equal bases mean equal exponents: $2x + 4 = 3x - 3$. Subtract $2x$ and add 3: $7 = x$, choice A. Check: $9^9 = 3^{18}$ and $27^6 = 3^{18}$. Choice B (9) is the base 9 and not a solution. Choice C (1) gives $9^3 = 729$ versus $27^0 = 1$. Choice D (3) gives $9^5 = 59049$ versus $27^2 = 729$. Distribute the 2 and 3 carefully, as a common error is $2x + 2 = 3x - 1$, which comes from forgetting to multiply the constants.