`(x^2 + 2x + 8)/(x^2 + 4)^2` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

`(x^2+2x+8)/[(x^2+4)^2]=(x^2+2x+8)/[(x^2+4)(x^2+4)]`

`(x^2+2x+8)/[(x^2+4)^2]=(Ax+B)/(x^2+4)+(Cx+D)/[(x^2+4)^2]`

Multiply through by the LCD `(x^2+4)^2.`

`x^2+2x+8=(Ax+B)(x^2+4)+(Cx+D)`

`x^2+2x+8=Ax^3+Bx^2+4Ax+4B+Cx+D`

`x^2+2x+8=Ax^3+Bx^2+(4A+C)x+(4B+D)`

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

`0=A`

`1=B`

`2=4A+C`

`2=4(0)+C`

`2=C`

`8=4B+D`

`8=4(1)+D`

`4=D`

`A=0, B=1, C=2, D=4`

`(x^2+2x+8)/[(x^2+4)^2]=(0x+1)/(x^2+4)+(2x+4)/[(x^2+4)^2]=1/(x^2+4)+(2x+4)/[(x^2+4)^2]`