Solve this

$\Delta=\left|\begin{array}{ccc}0 & b^{2} a & c^{2} a \\ a^{2} b & 0 & c^{2} b \\ a^{2} c & b^{2} c & 0\end{array}\right|$

$=\frac{1}{a b c}\left|\begin{array}{ccc}0 & b^{3} a & c^{3} a \\ a^{3} b & 0 & c^{3} b \\ a^{3} c & b^{3} c & 0\end{array}\right|$

[Multiplying the three columns by $a, b$ and $c$ ]

$=\frac{a b c}{a b c}\left|\begin{array}{ccc}0 & b^{3} & c^{3} \\ a^{3} & 0 & c^{3} \\ a^{3} & b^{3} & 0\end{array}\right|$

[Taking out $a, b$ and $c$ common from the three rows]

$=b^{3}\left|\begin{array}{cc}a^{3} & c^{3} \\ a^{3} & 0\end{array}\right|+c^{3}\left|\begin{array}{cc}a^{3} & 0 \\ a^{3} & b^{3}\end{array}\right|=2 a^{3} b^{3} c^{3} \quad\left[\right.$ Expanding along $\left.R_{1}\right]$