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

$\begin{array}{l}{\mathrm{x}}_1{\left(\mathrm{x}-{\mathrm{x}}_2\right)}^2+{\mathrm{x}}_2{\left(\mathrm{x}-{\mathrm{x}}_1\right)}^2=0\\ \Rightarrow {\mathrm{x}}^2\left({\mathrm{x}}_1+{\mathrm{x}}_2\right)-4{\mathrm{xx}}_1{\mathrm{x}}_2+{\mathrm{x}}_1{\mathrm{x}}_2\left({\mathrm{x}}_1+{\mathrm{x}}_2\right)=0\end{array}$

$\mathrm{D}=16{\left({\mathrm{x}}_1{\mathrm{x}}_2\right)}^2-4{\mathrm{x}}_1{\mathrm{x}}_2{\left({\mathrm{x}}_1+{\mathrm{x}}_2\right)}^2>0\left(\because {\mathrm{x}}_1{\mathrm{x}}_2<0\right)$

The product of roots is x₁x₂ < 0. Thus, the roots are real and of opposite signs.