`(x^2 + 12x + 12)/(x^3 - 4x)` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

`[x^2+12x+12]/[x^3-4x]=[x^2+12x+12]/[x(x+2)(x-2)]` 

`[x^2+12x+12]/(x^3-4x)=A/x+B/(x+2)+C/(x-2)`

Multiply through by the LCD `x^3+4x.`

`x^2+12x+12=A(x^2-4)+Bx(x-2)+Cx(x+2)`

`x^2+12x+12=Ax^2-4A+Bx^2-2Bx+Cx^2+2Cx`

`x^2+12x+12=(A+B+C)x^2+(-2B+2C)x+(-4A)`

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

`1=A+B+C`

`12=-2B+2C`

`12=-4A`

`A=-3`

`1=A+B+C`

`1=-3+B+C`

`4=B+C`

Solve for B and C using the elimination method.

`4=B+C`

`12=-2B+2C`

Multiply the first equation by 2. Then solve using the elimination method.

`8=2B+2C`

`12=-2B+2C`

_________________

20=4C

`C=5`

`1=A+B+C`

`1=-3+B+5`

`B=-1`

`A=-3, B=-1, C=5`

`(x^2+12x+12)/(x^3-4x)=-3/x-1/(x+2)+5/(x-2)`