Factorise
` x^(2) + (1)/(x^(2)) + 2- 5x -( 5)/(x)`

Answer

To factorize the expression \( x^2 + \frac{1}{x^2} + 2 - 5x - \frac{5}{x} \), we will follow these steps:

### Step 1: Rewrite the Expression
Start by rewriting the expression for clarity:
\[ x^2 + \frac{1}{x^2} + 2 - 5x - \frac{5}{x} \]

### Step 2: Combine Terms
Notice that \( x^2 + \frac{1}{x^2} + 2 \) can be rewritten using the identity \( a^2 + b^2 + 2ab = (a + b)^2 \). Here, let \( a = x \) and \( b = \frac{1}{x} \):
\[ x^2 + 2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 \]
Thus, we can rewrite the expression as:
\[ \left(x + \frac{1}{x}\right)^2 - 5x - \frac{5}{x} \]

### Step 3: Factor Out Common Terms
Now, we can express \( -5x - \frac{5}{x} \) as:
\[ -5\left(x + \frac{1}{x}\right) \]
So, the expression becomes:
\[ \left(x + \frac{1}{x}\right)^2 - 5\left(x + \frac{1}{x}\right) \]

### Step 4: Let \( y = x + \frac{1}{x} \)
Let \( y = x + \frac{1}{x} \). Then, we can rewrite the expression as:
\[ y^2 - 5y \]

### Step 5: Factor the Quadratic Expression
Now, we can factor the quadratic expression:
\[ y^2 - 5y = y(y - 5) \]

### Step 6: Substitute Back for \( y \)
Substituting back \( y = x + \frac{1}{x} \):
\[ (x + \frac{1}{x})(x + \frac{1}{x} - 5) \]

### Final Factorized Form
Thus, the factorized form of the expression is:
\[ (x + \frac{1}{x})(x + \frac{1}{x} - 5) \]