The roots of the equation are α,β,γ. So, α+β+γ=−p αβ+βγ+γα=q αβγ=−r Now, (1+α2)(1+β2)(1+γ2)=(1+α2)(1+β2+γ2+(βγ)2) =[1+β2+γ2+(βγ)2+α2+(αβ)2+(αγ)2+(αβγ)2] =[1+(α2+β2+γ2)+((βγ)2+(αβ)2+(αγ)2)+(αβγ)2] =[1+[(α+β+γ)2−2(αβ+βγ+γα)]+][(αβ+βγ+γα)2−2αβγ(α+β+γ)]] Simplify further, (1+α2)(1+β2)(1+γ2)=1+[p2−2q]+[q2−2rp]+r2 =1+p2−2q+q2−2rp+r2 =(q−1)2+(r−p)2