Prove that:
$\left|\begin{array}{ccc}1 & b+c & b^{2}+c^{2} \\ 1 & c+a & c^{2}+a^{2} \\ 1 & a+b & a^{2}+b^{2}\end{array}\right|=(a-b)(b-c)(c-a)$
Let LHS $=\Delta=\mid \begin{array}{lll}1 & b+c & b^{2}+c^{2}\end{array}$
$\begin{array}{lcc}1 & c+a & c^{2}+a^{2}\end{array}$
$\begin{array}{lll}1 & a+b & a^{2}+b^{2}\end{array}$
$\left[\right.$ Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$ and $\left.\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\right]$
$=\mid \begin{array}{lll}0 & b-a & b^{2}-a^{2}\end{array}$
$0 \quad c-b \quad c^{2}-b^{2}$
$1 \quad a+b \quad a^{2}+b^{2}$
$=(-1)^{2} \quad \mid 0 \quad a-b \quad a^{2}-b^{2}$
$\begin{array}{lll}0 & b-c & b^{2}-c^{2} \\ 1 & a+b & a^{2}+b^{2}\end{array}$
$\left[\right.$ Taking out $(-1)$ common from $\mathrm{R}_{1}$ and $\left.\mathrm{R}_{2}\right]$
$=(a-b)(b-c) \quad \mid 0 \quad 1 \quad a+b$
$\begin{array}{lll}0 & 1 & b+c \\ 1 & a+b & a^{2}+b^{2}\end{array}$
$=(a-b)(b-c)\{1 \times \mid 1 \quad a+b$
$1 \quad \mathrm{~b}+\mathrm{c} \mid\} \quad$ [Expanding along $\mathrm{C}_{1}$ ]
$=(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})$
$=\mathrm{RHS}$