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

1. Problem Description

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

2. Solution Steps

We can rewrite the expression by factoring out $3^{x-2}$:

$3^{x+2} - 6 \times 3^x + 5 \times 3^{x-2} = 3^4 \times 3^{x-2} - 6 \times 3^2 \times 3^{x-2} + 5 \times 3^{x-2}$.

Factoring out $3^{x-2}$, we have

$3^{x-2} (3^4 - 6 \times 3^2 + 5)$.

Now we evaluate the expression inside the parentheses:

$3^4 = 81$ and $3^2 = 9$.

So, we have

$3^{x-2} (81 - 6 \times 9 + 5) = 3^{x-2} (81 - 54 + 5)$.

Calculating the expression inside the parentheses, we get

$81 - 54 + 5 = 27 + 5 = 32$.

Therefore, the expression becomes

$32 \times 3^{x-2}$.

3. Final Answer

$32 \times 3^{x-2}$