We are asked to simplify the expression $3^{x+2} - 6 \cdot 3^x + 5 \cdot 3^{x-2}$.

1. Problem Description

We are asked to simplify the expression $3^{x+2} - 6 \cdot 3^x + 5 \cdot 3^{x-2}$.

2. Solution Steps

We can rewrite the expression using the properties of exponents.

$a^{m+n} = a^m \cdot a^n$

$a^{m-n} = \frac{a^m}{a^n}$

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

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

The expression becomes:

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

Now, let $y = 3^x$. The expression is now:

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

Combining like terms:

$(9 - 6 + \frac{5}{9})y = (3 + \frac{5}{9})y = (\frac{27}{9} + \frac{5}{9})y = \frac{32}{9}y$

Substitute $y = 3^x$ back in:

$\frac{32}{9} \cdot 3^x$

3. Final Answer

$\frac{32}{9} \cdot 3^x$