`A_6 = ( x/(x^2- 2x+1) - 1/(x^2 -1) ) . (2x- 2x^3)/(x^2+1) + 2/(x-1) (x \ne +-1)`
` = (x / (x-1)^2 - 1/((x-1)(x+1)) ) . (-2x (x^2 -1))/(x^2+1) +2/(x-1)`
` = ( (x (x+1))/( (x-1)^2 (x+1)) - (1 (x-1))/((x-1)^2 (x+1)) ) . (-2x (x-1)(x+1))/(x^2 +1) + 2/(x-1)`
` = ( x (x+1) - 1(x-1))/((x-1)^2(x+1)) . (-2x (x-1)(x+1))/(x^2+ 1) +2/(x-1)`
` = (x^2 + x - x + 1)/((x-1)^2(x+1)) . (-2x(x-1)(x+1))/(x^2+1) + 2/(x-1)`
` = (x^2+1)/((x-1)^2(x+1)) . (-2x (x-1)(x+1))/(x^2+1) + 2/(x-1)`
` = (-2x)/(x-1) + 2/(x-1)`
`= (-2x + 2)/(x-1)`
` = (-2 (x-1))/(x-1)`
`= -2`
``
`A_7 = (x+2)/(x-1) . ( (x^3)/(x+1) + 2) - (8x + 7)/(x^2 -1) (x \ne +-1)`
` = (x+2)/(x-1) . ( (x^3)/(x+1) + (2(x+1))/(x+1) ) - (8x + 7)/(x^2-1)`
` = (x+2)/(x-1) . ( x^3 + 2(x+1))/(x+1) - (8x+7)/(x^2-1)`
` = (x+2)/(x-1) . (x^3 + 2x + 2)/(x+1) - (8x+7)/(x^2-1)`
`= ((x+2)(x^3 + 2x+2))/(x^2-1) - (8x+7)/(x^2-1)`
`= ( (x+2)(x^3 + 2x+2) - (8x+7) ) /(x^2-1)`
` = ( x^4 + 2x^2 + 2x + 2x^3 + 4x + 4 - 8x - 7)/(x^2 - 1)`
` = (x^4 + 2x^3 + 2x^2 -2x -3)/(x^2 - 1)`
` = ( (x^4 - x^2) + (2x^3 - 2x) + (3x^2 - 3))/(x^2- 1)`
`= ( x^2 (x^2 -1) + 2x (x^2 - 1) + 3 (x^2 - 1))/(x^2- 1)`
` = ((x^2 + 2x + 3) (x^2-1))/(x^2 - 1)`
` = x^2 + 2x + 3`
Vậy `A_7 = x^2 + 2x + 3`
