Simplify the fraction (x^4+x)/(x^3-x).

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)].