Let $\Delta=\left|\begin{array}{lll}\mathrm{x} & \mathrm{x}^2 & \mathrm{yx} \\ \mathrm{y} & \mathrm{y}^2 & \mathrm{zx} \\ \mathrm{z} & \mathrm{z}^2 & \mathrm{xy}\end{array}\right|$
Applying $\mathrm{R}_1 \rightarrow \mathrm{R}_1-\mathrm{R}_2, \mathrm{R}_2 \rightarrow \mathrm{R}_2-\mathrm{R}_3$
$\begin{aligned}
\Delta &=\left|\begin{array}{ccc}
x-y & x^2-y^2 & y z-z x \\
y-z & y^2-z^2 & z x-x y \\
z & z^2 & x y
\end{array}\right| \\
&=\left|\begin{array}{ccc}
x-y & (x+y)(x-y) & z(y-x) \\
y-z & (y-z)(y+z) & x(z-y) \\
z & z^2 & x y
\end{array}\right|
\end{aligned}$
Taking common $(x-y),(y-z)$ from $R_1 \& R_2$ $\Delta=(x-y)(y-z)\left|\begin{array}{ccc}1 & x+y & -z \\ 1 & y+z & -x \\ z & z^2 & x y\end{array}\right|$
Operating $\mathrm{R}_1 \rightarrow \mathrm{R}_1-\mathrm{R}_2$
$\begin{aligned} \Delta=&(x-y)(y-z)\left|\begin{array}{ccc}0 & x-z & -z+x \\ 1 & y+z & -x \\ z & z^2 & x y\end{array}\right| \\=&(x-y)(y-z)(z-x)\left[0 \cdot\left\{(y+z) x y+x z^2\right\}\right] \\ &\left.\quad+1 \cdot(x y+x z)-\left(z^2-y z-z^2\right)\right] \\=&(x-y(y-z)(z-x)(x y+y z+z x)\\=& \text { R.H.S. } \end{aligned}$