Simple question about power

Question

Please solve this question and describe it. $$a^2+b^2+ab=b-a-1\Longrightarrow a^{1001}+b^{1001}=?$$ By Newton formula.

Answer 1

$a^2+b^2+ab=b-a-1 \implies (a+b)^2+(a+1)^2+(b-1)^2=0 \implies a=-1,b=1$