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\text{Now }xy^2+xz^2+yx^2+yz^2+zx^2+zy^2 \\[br]= y(xy+yz)+z(zx+yz)+x(xy+zx) \\[br]= y(xy+yz+zx)+z(xy+yz+zx)+x(xy+yz+zx) - 3xyz \\[br]= \alpha \beta - 3\gamma
\text{Then }16\Delta = \alpha(\alpha -2a)(\alpha-2b)(\alpha-2c)\\[br]=\alpha(\alpha^3 - 2(a+b+c)\alpha^2+4(ab+bc+ca)\alpha-8abc)[br]=\alpha(4\beta \alpha-\alpha^3-8\gamma).
\text{Now }xy^2+xz^2+yx^2+yz^2+zx^2+zy^2 \\[br]= y(xy+yz)+z(zx+yz)+x(xy+zx) \\[br]= y(xy+yz+zx)+z(xy+yz+zx)+x(xy+yz+zx) - 3xyz \\[br]= \alpha \beta - 3\gamma
\text{Then }16\Delta = \alpha(\alpha -2a)(\alpha-2b)(\alpha-2c)\\[br]=\alpha(\alpha^3 - 2(a+b+c)\alpha^2+4(ab+bc+ca)\alpha-8abc)[br]=\alpha(4\beta \alpha-\alpha^3-8\gamma).
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