Concept:Use elementary row and column operations to simplify the determinant into a Vandermonde form.Explanation:Start with the given determinant:1−α1−β1−γα−α2β−β2γ−γ2α2β2γ2Apply C1→C1+C2+C3. The first column becomes (1,1,1)T:111α−α2β−β2γ−γ2α2β2γ2Then apply C2→C2+C3. The second column becomes (α,β,γ)T:111αβγα2β2γ2Perform R1→R1−R2 and R3→R3−R2:R1=(0,α−β,α2−β2), R3=(0,γ−β,γ2−β2), R2=(1,β,β2).Expand along the first column. Only element 1 at (2,1) contributes with sign (−1)2+1=−1:=−α−βγ−βα2−β2γ2−β2Factor each entry: α2−β2=(α−β)(α+β), γ2−β2=(γ−β)(γ+β). Take common factors:=−(α−β)(γ−β)[(γ+β)−(α+β)]Simplify inside: γ+β−α−β=γ−α.=−(α−β)(γ−β)(γ−α)Rewrite (γ−β)=−(β−γ) and (γ−α)=−(α−γ). The two negatives multiply to positive:=(α−β)(β−γ)(γ−α).Answer:Option B: (α−β)(β−γ)(γ−α)