If α,β  are the roots of x2+a x−b=0  and γ, δ  are the roots of x2+a x+b=0 ,  then (α−γ)(α−δ)(β−δ)(β−γ)=

The given quadratic equations are x2+a x−b=0 …… ( 1 ) x2+a x+b=0 …….. ( 2 )Given α,β  are the roots of  Eq ( 1 )If x=α , x=β Then α2+aα−b=0   and   β2+aβ−b=0            ⇒α2+aα=b   and   β2+aβ=b Given γ, δ  are the roots of  Eq ( 2 )Then sum of the roots γ+δ=−a ,Product of the roots γ δ=b Now (α−γ)(α−δ)(β−δ)(β−γ)= [α2−α(γ+δ)+γδ][β2−β(γ+δ)+γδ]                                          =[α2+aα+b][β2+aβ+b]  (∵ γ+δ=−a, γδ=b)                                          =(b+b)(b+b)    (∵ α2+aα=b,β2+aβ=b)                                          =4b2