Let $\Delta=\left|\begin{array}{ccc}\mathrm{a}^2 & \mathrm{bc} & \mathrm{ac}+\mathrm{c}^2 \\ \mathrm{a}^2+\mathrm{ab} & \mathrm{b}^2 & \mathrm{ac} \\ \mathrm{ab} & \mathrm{b}^2+\mathrm{bc} & \mathrm{c}^2\end{array}\right|$
Taking common from $\mathrm{C}_1, \mathrm{~b}$ from $\mathrm{C}_2$ and $\mathrm{c}$ from $\mathrm{C}_3$. $\therefore \Delta=a b c\left|\begin{array}{ccc}a & b c & a+c \\ a+b & b & a \\ b & b+c & c\end{array}\right|$
Applying $\mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3$
$\Delta=\operatorname{abc}\left|\begin{array}{ccc}2(\mathrm{a}+\mathrm{c}) & \mathrm{c} & \mathrm{a}+\mathrm{c} \\ 2(\mathrm{a}+\mathrm{b}) & \mathrm{b} & \mathrm{a} \\ 2(\mathrm{~b}+\mathrm{c}) & \mathrm{b}+\mathrm{c} & \mathrm{c}\end{array}\right|$
$=2 a b c\left|\begin{array}{ccc}
a+c & c & a+c \\
a+b & b & a \\
b+c & b+c & c
\end{array}\right|$
Applying $\mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_1, \mathrm{C}_3 \rightarrow \mathrm{C}_3-\mathrm{C}_1$
$\Delta=2 a b c\left|\begin{array}{ccc}
\mathrm{a}+\mathrm{c} & -\mathrm{a} & 0 \\
\mathrm{a}+\mathrm{b} & -\mathrm{a} & -\mathrm{b} \\
\mathrm{b}+\mathrm{c} & 0 & -\mathrm{b}
\end{array}\right|$
Taking $-\mathrm{a}$ and $-\mathrm{b}$ common from $\mathrm{C}_2$ and $\mathrm{C}_3$ respectively
$\Delta=2 a^2 b^2 c=\left[\begin{array}{ccc}
a+c & 1 & 0 \\
a+b & 1 & 1 \\
b+c & 0 & 1
\end{array}\right]$
Expand along $\mathrm{C}_3$
$\begin{aligned}
\Delta &=2 \mathrm{a}^2 \mathrm{~b}^2 \mathrm{c}[\mathrm{b}+\mathrm{c}+(\mathrm{a}+\mathrm{c})-(\mathrm{a}+\mathrm{b})] \\
&=2 \mathrm{a}^2 \mathrm{~b}^2 \mathrm{c} \cdot 2 \mathrm{c} \\
\Delta &=4 \mathrm{a}^2 \mathrm{~b}^2 \mathrm{c}^2
\end{aligned}$