Simplify
1) x + 1/ x - 1 + x^2 -1 / x + 1
2) 4x/x^2 - 1 - x + 1/x - 1
x + 1/ x - 1 + x^2 -1 / x + 1
x+(1/x)-1+x^2-(1/x)+1
x-1+x^2+1
x+x^2
However, assuming the usual sloppiness with parentheses, I suspect you meant
(x+1)/(x-1) + (x^2-1)/(x+1)
(x+1)/(x-1) + (x-1)(x+1)/(x+1)
(x+1)/(x-1) + (x-1)
(x+1)/(x-1) + (x-1)^2/(x-1)
(x+1 + (x-1)^2)/(x-1)
(x^2-x+2)/(x-1)
x + 2/(x-1)
4x/x^2 - 1 - x + 1/x - 1
4/x - 1 - x + 1/x - 1
5/x - x - 2
Or, use parentheses to do it another way.
You might end up with
(1-x)/(1+x)