factor and simplify: A.(x^3+2x^2-3x)/(2x^3+2x^2-4x) ; B.(x^2y-x^2)/(x^3-x^3y) ; C.(x^2+5x+4)/(x^2-4x-5)

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A. You need to factor and reduce the fraction `(x^3+2x^2-3x)/(2x^3+2x^2-4x)`  to lowest terms, hence you need to factor out `x`  to numerator and denominator such that:

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

Reducing by x yields:

`(x^2+2x-3)/(2x^2+2x-4)`

You need to use quadratic formula to factor numerator and denominator such that:

`x^2+2x-3 = 0 =gt x_(1,2) = (-2 +- sqrt(4 + 12))/2`

`x_1 = (-2+4)/2 =gt x_1 = 1`

`` `x_2 = (-2-4)/2 =gt x_2 = -3`

`` `x^2+2x-3 = (x-1)(x+3)`

`` `2x^2+2x-4 = 0 =gt x^2+x-2 = 0`

`x_(3,4) = (-1 +- sqrt(1+8))/2 =gt x_3 = (-1+3)/2 =gt x_3 = 1`

`x_4 = (-1-3)/2 = -2`

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

Hence, you may write the factored form of the fraction such that:

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

You may reduce by x-1 both numerator and denominator such that:

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

Hence, reducing the fraction to lowest terms yields: `(x^3+2x^2-3x)/(2x^3+2x^2-4x) = (x+3)/(2(x+2))`

B. You need to factor and reduce the fraction`(x^2y-x^2)/(x^3-x^3y)`  to lowest terms, hence you need to factor out `x^2`  to numerator and `x^3 ` to denominator such that:

`(x^2y-x^2)/(x^3-x^3y) = (x^2(y-1))/(x^3(1-y))`

Reducing by `x^2(y-1)`  yields:

`(x^2(y-1))/(x^3(1-y)) = 1/(-x)`

Hence, reducing the fraction to lowest terms yields`:(x^2y-x^2)/(x^3-x^3y) = -1/x`

C. You need to factor and reduce the fraction `(x^2+5x+4)/(x^2-4x-5)`  to lowest terms, hence you need to use quadratic formula to factor numerator and denominator such that:

`x^2+5x+4 = 0 =gt x_(1,2) = (-5+-sqrt(25-16))/2`

`x_1 = (-5+3)/2 = -1`

`` `x_2 = (-5-3)/2 = -4`

`x^2+5x+4 = (x+1)(x+4)`

`` `x^2-4x-5=0 =gt x_(3,4) = (4+-sqrt(16+20))/2`

`x_3 = (-4 + 6)/2 =gt x_3 = 1`

`` `x_4 =(-4- 6)/2 =gt x_4 = -5`

`` `x^2-4x-5 = (x-1)(x+5)`

Hence, since the factored form of fraction `(x+1)(x+4)/(x-1)(x+5)`  does not possess like factors it is impossible to be reduced.

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