x1 and x2 are the roots of ax2 + bx + c = 0 and x1x2 < 0. Roots of x1(x - x2)2 + x2(x - x1)2 = 0 are

x1x−x22+x2x−x12=0⇒ x2x1+x2−4xx1x2+x1x2x1+x2=0D=16x1x22−4x1x2x1+x22>0∵x1x2<0The product of roots is x1x2 < 0. Thus, the roots are real and of opposite signs.