If the roots α,β of the equation px2+qx+r=0 are real  and of opposite signs (where p,q,r are real coefficients),  then the roots of the equation α(x−β)2+β(x−α)2=0 are

The equation α(x−β)2+β(x−α)2=0 or     (α+β)x2−2x(αβ+αβ)+αβ(α+β)=0 or      (α+β)x2−4αβx+αβ(α+β)=0 Product of its roots =αβ(α+β)(α+β)=αβ<0 given Hence roots are real and of opposite signs.