Understanding the Polynomial
To factorise the polynomial \( x^2 + \frac{1}{x^2} - 2x - \frac{2}{x} + 2 \), we will first rewrite it in a more manageable form.

Step 1: Combine Terms
We start by rewriting the polynomial in a common format:
- \( x^2 - 2x + 2 + \frac{1}{x^2} - \frac{2}{x} \)
Next, we will focus on the two separate parts of the expression.

Step 2: Factor Quadratic Terms
Let’s factor the quadratic expression \( x^2 - 2x + 2 \):
- This can be viewed as \( (x - 1)^2 + 1 \), which shows it does not factor nicely into real numbers.
For the second part, \( \frac{1}{x^2} - \frac{2}{x} \):
- Factor out \( \frac{1}{x^2} \):
- \( \frac{1}{x^2}(1 - 2x) \)
This suggests we focus on \( 1 - 2x \).

Step 3: Combine and Factor
Now combine the two parts:
- The polynomial can be rearranged:
- \( (x^2 - 2x + 2) + \frac{1 - 2x}{x^2} \)
Notice that both parts involve \( 2 - 2x \).

Final Factorisation
The polynomial can be expressed as:
- \( (x - 1)^2 + 1 + \frac{1}{x^2}(1 - 2x) \)
This expression may not have simple rational roots, but the essential combination can be simplified to show underlying structure.

Conclusion
Thus, the polynomial \( x^2 + \frac{1}{x^2} - 2x - \frac{2}{x} + 2 \) exhibits a complex factorisation that may not yield simple factors but can be expressed in component forms for analysis.