Factorising the expression:

To factorise the given expression x^2 - 1/x^2 + 2 - 2x - 2/x, we can follow the steps mentioned below:

Step 1: Identify common factors:
In the given expression, we can identify the common factor as (x - 1/x) since both terms have (x - 1/x) as a common factor.

Step 2: Rewrite the expression:
By factoring out the common factor (x - 1/x), we can rewrite the expression as follows:

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

Step 3: Simplify the expression:
To simplify the expression further, we can distribute the terms within the parentheses:

(x^2 - 1) + 2(x - 1/x) - 2(x - 1/x)

Simplifying this further, we get:

x^2 - 1 + 2x - 2/x - 2x + 2/x

Step 4: Combine like terms:
Now, let's combine the like terms in the expression:

x^2 - 1 - 2x + 2x + 2/x - 2/x

Simplifying this further, we obtain:

x^2 - 1

Final Answer:
Therefore, the given expression x^2 - 1/x^2 + 2 - 2x - 2/x can be factorised as (x - 1/x)(x + 2) - 2(x - 1/x), which simplifies to x^2 - 1.