If `alpha,beta ` be the roots of the equation `3x^2+2x+1=0,` then find value of `((1-alpha)/(1+alpha))^3+((1-beta)/(1+beta))^3`

Explanation

Let `(1-alpha)/(1+alpha)=ximpliesa=(1-x)/(1+x)`
So replacing `x` by `(1-x)/(1+x)` in the given equation we get
`3((1-x)/(1+x))^(2)+2((1-x)/(1+x))|+1=0impliesx^(2)-2x+3=0` …….i
It is clear that `(1-alpha)/(1+alpha)` and `(1-beta)/(1+beta)` are the roots fo Eq. (i)
`:.((1-alpha)/(1+alpha))+((1-beta)/(1+beta))=2`..........ii
and `((1-alpha)/(1+alpha))((1-beta)/(1+beta))=3`.......iii
`:.((1-alpha)/(1+alpha))^(3)+((1-beta)/(1+beta))^(3)=((1-alpha)/(1+alpha) +(1-beta)/(1+beta))^(3)-3`
`((1-alpha)/(1+alpha))((1-beta)/(1+beta))((1-alpha)/(1+alpha)+(1-beta)/(1+beta))=2^(3)-3.3.2=8-18=-10`