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

To simplify the expression \( \frac{1}{x} - 1 + \frac{1}{x} + x^2 + \frac{1}{x^2} - 1 \), we follow these steps:

1. Combine similar terms:
- The terms \( \frac{1}{x} \) appear twice, so we can add those together:
\( \frac{1}{x} + \frac{1}{x} = \frac{2}{x} \).
- The constants \( -1 - 1 \) can be combined: \( -1 - 1 = -2 \).
- The term \( x^2 \) and \( \frac{1}{x^2} \) remain unchanged.

2. Writing the combined equation, we have:
- \( \frac{2}{x} + x^2 + \frac{1}{x^2} - 2 \).

3. The expression now looks like:
- \( x^2 + \frac{1}{x^2} + \frac{2}{x} - 2 \).

4. We can further analyze \( x^2 + \frac{1}{x^2} \). There is a useful identity:
- \( x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \).
- If we set \( y = x + \frac{1}{x} \), then \( x^2 + \frac{1}{x^2} = y^2 - 2 \).

5. Now substituting that back into the expression:
- \( (y^2 - 2) + \frac{2}{x} - 2 \).

6. Simplifying:
- We now have \( y^2 - 4 + \frac{2}{x} \).

In conclusion, the simplified form of the original expression \( \frac{1}{x} - 1 + \frac{1}{x} + x^2 + \frac{1}{x^2} - 1 \) is \( x^2 + \frac{1}{x^2} + \frac{2}{x} - 2 \). This represents a combination of polynomial and rational terms, reflecting the interaction between \( x \) and \( \frac{1}{x} \).