Concept:Use the identity x3−1=(x−1)(x2+x+1) to simplify high powers of the roots.Explanation:Since α and β are roots of x2+x+1=0, they are the non-real cube roots of unity.From x3−1=(x−1)(x2+x+1), we get:α3=1,β3=1Also, by Vieta's formulas,α+β=−11=−1Now 2026=3×675+1.Therefore,α2026=α3×675+1=(α3)675⋅α=1⋅α=αSimilarly,β2026=βAdding:α2026+β2026=α+β=−1Answer:−1Option A.