Question

Factorize x Square +1/x Square +2x-2/x

Answer 1

Given equation is incorrect. There can be no exact answer of the given Quadratic Equation. 
Correct equation will be,
 x² + 1/x² - 2x - 2/x + 2

Now, Solving this,
 x² + 1/x² + 2 - 2x - 2/x
= [x² + 1/x² + 2] - 2[x + 1/x]
= [x + 1/x]² - 2[x + 1/x]
(∵ [x² + 1/x² + 2] can be written as [x + 1/x]²)
= [x + 1/x][ x + 1/x - 2]


The Required solution of the above Quadratic Equation will be ⇒
 [x + 1/x][ x + 1/x - 2]


Hope it helps.

Answer 2

x²+1/x²+2x-2/x
(x²+1/x²-2) + (x-1/x) + 2
(x-1/x)² + (x-1/x) + 2
(x-1/x)(x-1/x+1)+2
The given equation cannot be factorized any further.

If the equation was x²-1/x²+2x-2/x, it could be factorized.
(x-1/x)(x+1/x) + 2(x-1/x)
(x-1/x)(x+1/x+2)