Precalculus, Chapter 7, 7.4, Section 7.4, Problem 30

(6x^2+1)/[x^2(x-1)^2]=A/x+B/x^2+C/(x-1)+D/(x-1)^2
Multiply by the LCD x^2(x-1)^2.
6x^2+1=Ax(x-1)^2+B(x-1)^2+Cx^2(x-1)+Dx^2
6x^2+1=Ax(x^2-2x+1)+B(x^2-2x+1)+Cx^3-Cx^2+Dx^2
6x^2+1=Ax^3-2Ax^2+Ax+Bx^2-2Bx+B+Cx^3-Cx^2+Dx^2
6x^2+1=(A+C)x^3+(-2A+B-C+D)x^2+(A-2B)x+B

Equate coefficients of like terms. Then solve for A, B, C, and D.
B=1

0=A-2B
0=A-2(1)
A=2

0=A+C
0=2+C
C=-2

6=-2A+B-C+D
6=-2(2)+1-(-2)+D
6=-4+1+2+D
D=7

A=2, B=1, C=-2, D=7

(6x^2+1)/[x^2(x-1)^2]=2/x+1/x^2+[-2/(x-1)]+7/(x-1)^2