We are asked to solve the equation $3^{x+2} - 6 \cdot 3^x + 5 \cdot 3^{x-2} = 0$ for $x$.

1. Problem Description

We are asked to solve the equation $3^{x+2} - 6 \cdot 3^x + 5 \cdot 3^{x-2} = 0$ for $x$.

2. Solution Steps

First, we can rewrite the terms in the equation using the properties of exponents:

$3^{x+2} = 3^x \cdot 3^2 = 9 \cdot 3^x$

$3^{x-2} = 3^x \cdot 3^{-2} = \frac{1}{9} \cdot 3^x$

Substituting these back into the equation, we get:

$9 \cdot 3^x - 6 \cdot 3^x + 5 \cdot \frac{1}{9} \cdot 3^x = 0$

Now, let $y = 3^x$. Then the equation becomes:

$9y - 6y + \frac{5}{9}y = 0$

Combining the terms, we have:

$3y + \frac{5}{9}y = 0$

$\frac{27y + 5y}{9} = 0$

$\frac{32y}{9} = 0$

$32y = 0$

$y = 0$

Since $y = 3^x$, we have $3^x = 0$.

However, $3^x$ is always positive for any real number $x$, so $3^x$ can never be equal to $0$. Therefore, there is no solution to the equation.

3. Final Answer

There is no solution.