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

1. Problem Description

We need 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 using exponent rules:

$a^{b+c} = a^b \cdot a^c$ and $a^{b-c} = \frac{a^b}{a^c}$

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

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

Substituting these back into the equation:

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

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

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

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

Multiplying the entire equation by 9 to eliminate the fraction:

$27y + 5y = 0$

$32y = 0$

$y = 0$

Since $y = 3^x$, we have $3^x = 0$. However, $3^x$ can never be equal to 0 for any real number $x$.

Therefore, there is no solution.

3. Final Answer

No solution.