Foctorise :
` x^(2) +(1)/(x^(2)) - 2 -3 x+ (3)/(x)`

Answer

To factorize the expression \( x^2 + \frac{1}{x^2} - 2 - 3x + \frac{3}{x} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^2 + \frac{1}{x^2} - 2 - 3x + \frac{3}{x} \]

### Step 2: Combine like terms
Rearranging the expression, we can group the terms:
\[ (x^2 - 3x) + \left(\frac{1}{x^2} + \frac{3}{x} - 2\right) \]

### Step 3: Recognize a perfect square
Notice that \( x^2 - 3x \) can be rewritten using the formula for a perfect square:
\[ x^2 - 3x = (x - \frac{3}{2})^2 - \frac{9}{4} \]
However, we will focus on the term \( x^2 + \frac{1}{x^2} - 2 \) first.

### Step 4: Use the identity for squares
The expression \( x^2 + \frac{1}{x^2} - 2 \) can be recognized as:
\[ (x - \frac{1}{x})^2 \]
So we can rewrite the expression as:
\[ (x - \frac{1}{x})^2 - 3x + \frac{3}{x} \]

### Step 5: Substitute \( t = x - \frac{1}{x} \)
Let \( t = x - \frac{1}{x} \). Then we have:
\[ t^2 - 3t \]

### Step 6: Factor out \( t \)
Now, we can factor out \( t \):
\[ t(t - 3) \]

### Step 7: Substitute back for \( t \)
Substituting back \( t = x - \frac{1}{x} \):
\[ (x - \frac{1}{x})(x - \frac{1}{x} - 3) \]

### Final Factorized Form
Thus, the final factorized form of the expression is:
\[ (x - \frac{1}{x})(x - \frac{1}{x} - 3) \]