To simplify the expression, let's start by simplifying each term separately.

1. Simplify the numerator:

căn x / căn x - 2 + 2 / x - 2căn x

= căn x / căn x - 2 + 2 / (x - 2) * căn x

= căn x / căn x - 2 + 2căn x / (x - 2)

= (căn x + 2căn x) / (căn x - 2)(x - 2)

= 3căn x / (căn x - 2)(x - 2)

2. Simplify the denominator:

2 / căn x - 2 + 1

= 2 / (căn x - 2) + 1

Now, let's simplify the whole expression by dividing the numerator by the denominator:

(3căn x / (căn x - 2)(x - 2)) / (2 / (căn x - 2) + 1)

To divide by a fraction, we multiply by its reciprocal:

(3căn x / (căn x - 2)(x - 2)) * ((căn x - 2) + 1) / 2

Simplifying further:

(3căn x * (căn x - 2 + 1)) / (2 * (căn x - 2)(x - 2))

= (3căn x * (căn x - 1)) / (2 * (căn x - 2)(x - 2))

= (3căn x * căn x - 3căn x) / (2 * (căn x - 2)(x - 2))

= (3x - 3căn x) / (2 * (căn x - 2)(x - 2))

Therefore, the simplified expression is (3x - 3căn x) / (2 * (căn x - 2)(x - 2)).