Precalculus, Chapter 7, 7.4, Section 7.4, Problem 52

(3x^2-7x-2)/(x^3-x)=(3x^2-7x-2)/[x(x+1)(x-1)]
(3x^2-7x-2)/(x^3-x)=A/x+B/(x+1)+C/(x-1)

Multiply through by the LDC x^3-x.
3x^2-7x-2=A(x^2-1)+Bx(x-1)+Cx(x+1)
3x^2-7x-2=Ax^2-A+Bx^2-Bx+Cx^2+Cx
3x^2-7x-2=(A+B+C)x^2+(-B+C)x-A

Equate the coefficients of like terms. Then solve for A, B, and C.
3=A+B+C
-7=-B+C
-2=-A

A=2

3=A+B+C
3=2+B+C
1=B+C

Use the elimination method to solve for B and C.
1=B+C
-7=-B+C
_______________
-6=2C
-3=C

1=B+C
1=B-3
4=B

A=2, B=4, C=-3

(3x^2-7x-2)/(x^3-x)=2/x+4/(x+1)-3/(x-1)