Given #(2/(x^2-1)-1/(x+1))/(1/(12x^2-3))#
Division by a fraction is the same as multiplication by its reciprocal:
#(2/(x^2-1)-1/(x+1))(12x^2-3)/1#
Use the distributive property:
#(2(12x^2-3))/(x^2-1)-(1(12x^2-3))/(x+1)#
Twice:
#(24x^2-6)/(x^2-1)+(-12x^2+3)/(x+1)#
Add #-24+24# to the first term and #-12x+12x# to the second term:
#(24x^2-24+24-6)/(x^2-1)+(-12x^2-12x+12x+3)/(x+1)#
Separate into 4 fractions:
#(24x^2-24)/(x^2-1)+(18)/(x^2-1)+(-12x^2-12x)/(x+1)+
(12x+3)/(x+1)#
Two of the fractions reduce:
#-12x+24+(18)/(x^2-1)+(12x+3)/(x+1)#
Add +12-12 to the second fraction:
#-12x+24+(18)/(x^2-1)+(12x+12-12+3)/(x+1)#
Separate the second fraction into two fractions:
#-12x+24+(18)/(x^2-1)+(12x+12)/(x+1)+(-9)/(x+1)#
One of the fractions reduces:
#-12x+24+(18)/(x^2-1)+12+(-9)/(x+1)#
Combine like terms:
#-12x+36+(18)/(x^2-1)+(-9)/(x+1)#
The fraction #(18)/(x^2-1)# decomposes:
#18/(x^2-1) = A/(x-1)+B/(x+1)#
#18 = A(x+1)+B(x-1)#
Let x = 1:
#18 = A(1+1)+B(1-1)#
#A = 9#
Let x = -1:
#B = -9#
#18/(x^2-1) = 9/(x-1)+(-9)/(x+1)#
Substitute back into the expression:
#-12x+36+9/(x-1)+(-9)/(x+1)+(-9)/(x+1)#
Combine like terms:
#-12x+36+9/(x-1)-18/(x+1)#
Done.
