Explanation

(a)
$\left| \begin{matrix}
1 & a & a^{2} - bc \\
1 & b & b^{2} - ac \\
1 & c & c^{2} - ab
\end{matrix} \right| = \left| \begin{matrix}
0 & a - b & (a - b)(a + b + c) \\
0 & b - c & (b - c)(a + b + c) \\
1 & c & c^{2} - ab
\end{matrix} \right|$ $\binom{R_{1} \rightarrow R_{1} - R_{2}}{R_{2} \rightarrow R_{2} - R_{3}}$
$= (a - b)(b - c)\left| \begin{matrix}
0 & 1 & a + b + c \\
0 & 1 & a + b + c \\
1 & c & c^{2} - ab
\end{matrix} \right| = 0$ $(\because rows\ R_{1}\ and\ R_{2}$ are identical)