Question 1135174

I done the following rational expression addition and subtraction 4*x/(x^2-1)+3*x/(1-x)-4/(x-1) and had some issues with it , I have seen the full answer and got stuck on how some things were arrived at . I wonder if some one would kindly shed some light on the following. 1, in  both the second term -3*x/1-x and third term  -4/(x-1) could a greater explanation of how or why the denominators are allocated to there numerators? I would have thought that the  third term -4/(x-1) would equal -4*(-x+1)*(x+1)/(x+1)*(-x+1)*(x-1) instead of -4*(x+1)*(x-1) ?  2, and finally an explanation of how factoring the total numerator is achieved ?  As all this would be much appreciated kind regards mike.
<pre>{{{4x/(x^2 - 1) + 3x/(1 - x)  -  4/(x - 1)}}}
{{{matrix(2,3, x^2 - 1, "=", (x - 1)(x + 1), (1 - x), "=", - (x - 1))}}}
The LCD is then: - (x + 1)(x - 1), and {{{4x/(x^2 - 1) + 3x/(1 - x) - 4/(x - 1)}}} becomes:
{{{(- 1(4x))/(- (x + 1)(x - 1)) + 3x(x + 1)/(- (x + 1)(x - 1)) - 4(- 1)(x + 1)/(- (x + 1)(x - 1))}}} ------- Multiplying EACH EXPRESSION by LCD, -(x + 1)(x - 1)
{{{4x/(x + 1)(x - 1) - 3x(x + 1)/(x + 1)(x - 1) + (- 4)(x + 1)/(x + 1)(x - 1)}}} <==== Same as above
{{{highlight_green(matrix(1,5, (4x - 3x(x + 1) - 4(x + 1))/((x + 1)(x - 1)), "=====>", (4x - 3x^2 - 3x - 4x - 4)/(x + 1)(x - 1), "======>", highlight(matrix(1,3, (- 3x^2 - 3x - 4)/(x + 1)(x - 1), or, (3x^2 + 3x + 4)/(- (x + 1)(x - 1))))))}}}