`(4x^2 + 2x - 1)/(x^2(x + 1))` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

`(4x^2+2x-1)/[x^2(x+1)]=A/x+B/x^2+C/(x+1)`

Multiply through by the LCD `x^2(x+1).`

`4x^2+2x-1=Ax(x+1)+B(x+1)+Cx^2`

`4x^2+2x-1=Ax^2+Ax+Bx+B+Cx^2`

`4x^2+2x-1=(A+C)x^2+(A+B)x+B`

Equate the coefficients of like terms. Then solve for A, B, and C.

`B=-1`

`2=A+B`

`2=A-1`

`A=3`

`4=A+C`

`4=3+C`

`C=1`

`A=3, B=-1, C=1`

`(4x^2+2x-1)/[x^2(x+1)]=3/x-1/x^2+1/(x+1)`