To find the length of one side of the square room, we need to determine the side length from the area given, which is \( 9x^2 - 6x + 1 \) square feet.
Since the area of a square room can be expressed as \( s^2 \), where \( s \) is the length of one side, we need to factor the area expression \( 9x^2 - 6x + 1 \).
Let's factor \( 9x^2 - 6x + 1 \). We can look for factors of the quadratic expression.
The expression \( 9x^2 - 6x + 1 \) is a perfect square trinomial:
\[ 9x^2 - 6x + 1 = (3x - 1)^2 \]
Now, to confirm this, we can expand \( (3x - 1)^2 \):
\[ (3x - 1)(3x - 1) = 9x^2 - 3x - 3x + 1 = 9x^2 - 6x + 1 \]
Thus, we have verified that \( 9x^2 - 6x + 1 = (3x - 1)^2 \).
Since the area of the room is \( (3x - 1)^2 \), the length of one side \( s \) is:
\[ s = 3x - 1 \]
Hence, the length of one side of the room is:
\[ \boxed{(3x - 1) \text{ feet}} \]