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The root of x^2+x-1=0 is (-1+sqrt5)/2, so plug it in and you get 2.

x^2 + x  -  1  =  0

Rearrange as  :

x^2  =  1 - x      (1)

And

x^4 - 3x^2 + 3   =  

(x^2)^2 - 3x^2 + 3     (2)

Sub (1)  into (2)     and we have that

(1 - x)^2 - 3 (1 - x) +3

x*2 - 2x + 1 - 3 + 3x + 3

x^2 + x  - 2 + 3

( x^2 + x - 1 )  - 1 + 3

( x^2 + x - 1 ) + 2

0  +  2   =

2