Explanation

(a)
Given, $\mathrm{\Delta} = \left| \begin{matrix}
1 & 1 & 1 \\
a & b & c \\
a^{3} & b^{3} & c^{3}
\end{matrix} \right| = \left| \begin{matrix}
1 & 0 & 0 \\
a & a - b & a - c \\
a^{3} & a^{3} - b^{3} & a^{3} - c^{3}
\end{matrix} \right|$
$\lbrack C_{2} \rightarrow C_{1} - C_{2},C_{3} \rightarrow C_{1} - C_{3}\rbrack$
$= (a - b)(a - c)\ \left| \begin{matrix}
1 & 0 & 0 \\
a & 1 & 1 \\
a^{3} & a^{2} + ab + b^{2} & a^{2} + ac + c^{2}
\end{matrix} \right|$
$= (a - b)(a - c)(c^{2} + ac - ab - b^{2})$
$= (a - b)(b - c)(c - a)(a + b + c)$