Solve the equation: 27^x + 144^x+1 = 81

The equation to solve is:

$$27^x + 144^{x+1} = 81$$

Step 1: Rewrite the bases in terms of their prime factors or common bases where possible.
We know that $27 = 3^3$ and $81 = 3^4$.
The number 144 can be written as $12^2$.
Substitute these into the equation:

$$(3^3)^x + (12^2)^{x+1} = 3^4$$ $$3^{3x} + 12^{2(x+1)} = 3^4$$ $$$$