Given the cubic of X + twice of X + X + 1 = 0, find the value of twice of 1 + X + X + thrice of X + --- + 2012 times of X

Given the cubic of X + twice of X + X + 1 = 0, find the value of twice of 1 + X + X + thrice of X + --- + 2012 times of X

1+x+x^2+x^3=0
1+x+x^2+x^3+...+x^2012
Starting from x ^ 4, there are 2012-4 + 1 = 2009 items, because 502 * 4 = 2008 can be divided into 2008 x ^ k (1 + X + x ^ 2 + x ^ 3) plus x ^ 2012
So the original formula = (1 + X + x ^ 2 + x ^ 3) + x ^ 4 (1 + x ^ + x ^ 2 + x ^ 3) +. X ^ k (1 + X + x ^ 2 + x ^ 3) + x ^ 2012
=0+0+.+x^2012
=x^2012
x^3+x^2+x+1=0
(x^2(x+1))+x+1=0
(x+1)(x^2+1)=0
x=-1
So the original formula = (- 1) ^ 2012 = 1

Given the quadratic minus x minus one of X, try to find the value of cubic minus two of x minus one

x^2-x-1=0
x^2-x=1
x^3-2x-1
=x(x^2-x)+(x^2-x)-x-1
=x+1-x-1
=0

Given twice + X-1 = 0 of X, find twice + 2004 of thrice + 2x of X
Math genius, come and have a look

Twice of X + X-1 = 0
Twice of x = 1-x
therefore
Three times of X + two times of 2x
=X^2(X+2)
=(1-X)(X+2)
=X+2-X^2-2X
=-X^2-X+2
=-1+2
=1
Three times of X + two times of 2x + 2004 = 2005