How to find $α^2(β^4 +γ^4 +δ^4)+β^2(γ^4 +δ^4 +α^4)+γ^2(δ^4 +α^4 +β^4)+δ^2(α^4 +β^4 +γ^4)$ 

How to do the part (iv) . Please help.


Here are my answers to the first parts: 
(i)
α a root of given equation $\implies \alpha^4-5 \alpha^2 + 2 \alpha -1 = 0$
$\implies \alpha^{n+4} - 5 \alpha^{n+2} + 2\alpha^{n+1} -\alpha^n=0$
Summing over $α, \beta , \gamma , \sigma$, leads to
$S_{n+4}– 5S_{n + 2} + 2S_{n +1}- S_n=0$
(ii) 
$S_2=10$
$S_4 = 5S_2 – 2S_1 + 4 = 50 – 0 + 4 = 54$
(iii)
$S_{-1} = 2$ from $y^4 – 2y^3+5y^2-1=0$
$S_3 = 5S_1 – 2S_0 + S_{–1} = –6$
$S_6 = 5S_4 – 2S_3 + S_2 = 292$

I have no clue how to (iv)

Note that $\beta^4 + \gamma^4 + \delta^4 = S_4 - \alpha^4$, and similarly for the other quantities in parentheses.  Substitute this in your desired expression, and expand it.  You can then write the entire expression in terms of various $S$'s.
