Question

How do you long divide `(x-3)/(x^3+2x^2+x)`?

Answer 1

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

Explanation:

I think you want partial fractions.
Factorize below: `x(x^2 + 2x + 1) = x(x + 1)^2`

`frac{x-3}{x^3 + 2x^2 + x} = A/x +B/(x + 1) + C/(x + 1)^2`
`exists A,B,C in RR`

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

`x^2 : 0 = A + B`
`x^1: 1 = 2A + B + C`
`x^0: -3 = A Rightarrow B = +3`

`1 = 2(-3) + 3 + C Rightarrow C = 4`