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[ 2x² + 2x - 2 ] / [ x(x² - 1) ]  =  [ 2x² + 2x - 2 ] / [ x(x + 1)(x - 1) ] 

 

To get each term to have the common denominator of x(x + 1)(x - 1),

--  multiply the first term:  A/x  by  [ (x + 1)(x - 1) ] / [ (x + 1)(x - 1)]

        so that the numerator becomes:  A(x + 1)(x - 1)  =  Ax² - A

--  multiply the second term:  B/(x - 1)  by  [ x(x + 1) ] / [ x(x + 1) ]

        so that the numerator becomes:  B(x)(x + 1)  =  Bx² + Bx

--  multiply the third term:  C/(x + 1)  by  [ x(x - 1) ] / [ x(x - 1) ]

        so that the numerator becomes:  C(x)(x - 1)  =  Cx² - Cx

 

All the denominator are  (x)(x + 1)(x - 1), so all we have to do is look at the numerators:

     2x² + 2x - 2  =   Ax² - A  +  Bx² + Bx  +  Cx² - Cx

     2x² + 2x - 2  =   Ax² +  Bx² +  Cx²  + BX - Cx - A 

 

Therefore:  Ax² +  Bx² +  Cx²  =  2x²      --->     (A + B + C)x²  =  2x²     --->     A + B + C  =  2

                                   Bx - Cx  =  2x       --->               (B - C)x  =  2x      --->            B - C  =  2

                                            -A  =  -2                                                        --->                  A  =  2

 

Solving, we get:  A = 2     B = 1     C = -1