`(x^4 + 2x^3 + 4x^2 + 8x + 2)/(x^3 + 2x^2 + x)` Write the partial fraction decomposition of the improper rational expression.

First we have to make the fraction a proper one, for this it is necessary to divide the numerator by the denominator with the remainder.

`x^4+2x^3+4x^2+8x+2 =x(x^3+2x^2+x) + 3x^2 +8x+2.`

So `(x^4 + 2x^3 + 4x^2 + 8x + 2)/(x^3 + 2x^2 + x) = x+(3x^2+8x+2)/(x^3 + 2x^2 + x)=x+(3x^2+8x+2)/(x(x+1)^2).`

The proper part has the decomposition form of

`(3x^2+8x+2)/(x(x+1)^2)=A/x+B/(x+1)+C/(x+1)^2.`

Multiply both sides by `x(x+1)^2` and obtain

`3x^2+8x+2=A(x^2+2x+1)+Bx(x+1)+Cx=`

`=x^2*(A+B)+x*(2A+B+C)+A.`

So `A+B=3,` `2A+B+C=8` and `A=2,` from this we get `B=1` and `C=3.`

The answer: `(x^4 + 2x^3 + 4x^2 + 8x + 2)/(x^3 + 2x^2 + x) = x+2/x+1/(x+1)+3/(x+1)^2.`