It is given that α and β are roots of the quadratic equation x2+px−q=0. Therefore,α+β=−pαβ=−q ........(1) Similarly, γ and delta are roots of the quadratic equation x2−px+r=0 Therefore, γ+δ=pγδ=r .... (2) Now consider the given expression: (β+γ)(β+δ)=β2+γβ+δβ+γδ =β2+β(γ+δ)+γδ =β2+β(p)+r Since β is a root of the equation x2+px−q=0, it satisfies that equation. Therefore, β2+pβ−q=0 that implies β2+pβ=q Therefore, the given expression becomes: (β+γ)(β+δ)=β2+pβ+r =q+r Hence, (β+γ)(β+δ)=q+r