`(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`

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