For any scalar `p` prove that `=|xx^2 1+p x^3y y^2 1+p y^3z z^2 1+p z^3|=(1+p x y z)(x-y)(y-z)(z-x)` .

Explanation

LHS $$=\left|\begin{array}{lll}x & x^{2} & 1+p x^{3} \\ y & y^{2} & 1+p y^{3} \\ z & 2^{2} & 1+p z^{3}\end{array}\right|$$
By splitting into two parts, we get
$$=\left|\begin{array}{llllll}x & x^{2} & 1 & x & x^{2} & p x^{3} \\ y & y^{2} & 1+y & y^{2} & p y^{3} \\ z & z^{2} & 1 & z & z^{2} & p z^{3}\end{array}\right|$$
In second determinant, replacing `c_{1}` and `c_{3}` and then `c_{1}` with `c_{2}`]
$$
=(1+\operatorname{pxy} y)\left|\begin{array}{lll}
x & x^{2} & 1 \\
y & y^{2} & 1 \\
...