We have to simplify: [4-(3+x)/x]/[(x-1)/(x+1)]-[4/(x-x^2)]

[4-(3+x)/x]/[(x-1)/(x+1)]-[4/(x-x^2)]

=> [4(x + 1)/x(x - 1)] - 4/(x - x^2)

=> [(4x + 4)/(x^2 - x)] - 4/(x - x^2)

=> (-4x - 4)/(x - x^2) - 4/(x - x^2)

=> (-4x - 4 - 4)/(x - x^2)

=> (-4x - 8)/(x - x^2)

**The simplified expression is (-4x - 8)/(x - x^2)**

For the first term of the expression:

[4-(3+x)/x] /[ (x-1)/(x+1)]

The numerator of the first term is 4-(3+x)/x and the denominator is (x-1)/(x+1). To write the numerator in a simplified manner, we have to have the same denominator, which is x. For this reason, we'll multiply 4 by x.

The numerator will become:

(4x-3-x)/x=(3x-3)/x=3(x-1)/x

The numerator 3(x-1)/x will be multiplied by the inversed ratio(x+1)/(x-1).

[3(x-1)/x]*[(x+1)/(x-1)],

We'll simplify the common factor (x-1),and we'll get:

3(x+1)/x

This result will be added to the second term -[ 4/(x-2x^2)]

[3(x+1)/x] - [ 4/(x-2x^2)]

[3(x+1)/x]-[ 4/x(1-2x)]

We've noticed that the LCD of the 2 ratios is

x(1-2x), so we'll multiply the first ratio by (1-2x).

The expression will be:

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

We'll combine like terms:

E(x)=(-3x-6x^2-1)/x(1-2x)

We'll re-arrange the terms:

E(x)=(6x^2+3x+1)/x(1-2x)