This question seems to be a bit more complicated than what is normally in an FP1 paper, so don't worry too much. May I suggest using the "finding the new roots method", as you were doing previously. So we need to find the sum of the roots, the sum of the product of pairs of roots, and the product of all the roots.
Sum of roots =
γβ+βγ+αγ+γα+βα+αβ and you got to
αβγα(β2+γ2)+β(α2+γ2)+γ(α2+β2).
Note that this simplifies further to
Unparseable latex formula: \displaystyle\frac {(\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha) - 3\alpha\beta\gamma}{\alphga\beta\gamma}
(You might want to check to see if that's correct :P ). You know the values of all of the things in that fraction from the above equations, so you can obtain the new sum of the roots.
Now you will need to repeat this for the product of pairs and the product of the roots.