Answer
See explanations.
Work Step by Step
Step 1. Start from the original, do the operations: $R_1-R_2\to R_1$ and $R_2-R_3\to R_2$
$\begin{array}( \\|A|= \\ \\ \end{array}
\begin{vmatrix} 0&x-y&x^2-y^2\\0&y-z&y^2-z^2\\1&z&z^2 \end{vmatrix}$
Step 2. Use the last row for expansion:
$\begin{array}( \\|A|= \\ \\ \end{array}
\begin{vmatrix} x-y&x^2-y^2\\y-z&y^2-z^2\end{vmatrix}$
Step 3. Complete the expansion:
$|A|=(x-y)(y-z)(y+z)-(x-y)(x+y)(y-z)=(x-y)(y-z)(y+z-x-y)$
Step 4. We proved the expression:
$|A|=(x-y)(y-z)(z-x)$
