m2+m-1=0, n2-n-1=0代数式m3+n3+2m2-2n2+2008的值

m2+m-1=0, n2-n-1=0代数式m3+n3+2m2-2n2+2008的值
m2+m-1=0, n2-n-1=0
m^2+m=1,n^2-n=1
m3+n3+2m2-2n2+2008
=m^3+m^2+m^3-m^2+m^2-m^2+2008
=m(m^2+m)+n(n^2-n)+m^2-n^2+2008
=m+n+m^2-n^2+2008
=(m^2+m)-(n^2-n)+2008
=1-1+2008
=2008

m^2+m=1,n^2-n=1

m^3+n^3+2m^2-2n^2+2008
=m(m^2+m)+n(n^2-n)+m^2-n^2+2008
=m+n+m^2-n^2+2008
=(m^2+m)-(n^2-n)+2008
=1-1+2008
=2008

m^3+m^2-m=0;
n^3+n^2-n=0;
m^3+n^3+2m^2-2n^2+2008=
m^2-n^2+m+n=
(m^2+m-1)-(n^2-n-1)+1-1+2008=
0-0+1-1+2008=
2008

m^2+m-1=0
n^2-n-1=0
(-n)^2+(-n)-1=0
所以
若m=-n
则两式可以成立
则m^3+n^3+2m^2-2n^2+2008
=(-n)^3+n^3+2(-n)^2-2n^2+2008
=-n^3+n^3+2n^2-2n^2+2008
=2008

若m不等于-n
则m和-n是方程x^2+x-1=0的两个根
则m+(-n)=-1,m*(-n)=-1
m-n=-1,mn=1
m^2+n^2=(m-n)^2+2mn=1+2=3
(m+n)^2=m^2+2mn+n^2=3+2=5
所以m+n=±√5
m^3+n^3+2m^2-2n^2+2008
=(m+n)(m^2-mn+n