We have to simplify (x^4 + x)/(x^3 - x)

(x^4 + x)/(x^3 - x)

=> x(x^3 + 1)/x(x^2 - 1)

=> (x^3 + 1)/(x^2 - 1)

=> (x + 1)(x^2 - x + 1)/(x - 1)(x + 1)

=> (x^2 - x + 1)/(x - 1)

**The required result is (x^2 - x + 1)/(x - 1)**

To simplify the given fraction, we must factorize by x both, numerator and denominator.

(x^4+x)/(x^3-x) = x(x^3 + 1)/x(x^2 - 1)

We'll simplify and we'll get:

x(x^3 + 1)/x(x^2 - 1) = (x^3 + 1)/(x^2 - 1)

We notice that the numerator is a sum of cubes:

x^3 + 1 = (x+1)(x^2 - x + 1)

We notice that the denominator is a difference of squares:

x^2 - 1 = (x-1)(x+1)

We'll re-write the fraction:

(x^3 + 1)/(x^2 - 1) = (x+1)(x^2 - x + 1)/(x-1)(x+1)

We'll reduce by (x+1):

(x+1)(x^2 - x + 1)/(x-1)(x+1) = (x^2 - x + 1)/(x-1)

**The given simplified fraction is: (x^4+x)/(x^3-x) = x + [1/(x-1)**].