Let $\Delta=\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|$

Applying $\quad \mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}-\mathrm{R}_{3}$ to $\Delta,$ we get

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

Expanding along $\mathrm{R}_{1},$ we obtain

$\Delta  = 0\left| {\begin{array}{*{20}{c}}
  {c + a}&b \\ 
  c&{a + b} 
\end{array}} \right| – ( – 2c)\left| {\begin{array}{*{20}{c}}
  b&b \\ 
  c&{a + b} 
\end{array}} \right| + ( – 2b)\left| {\begin{array}{*{20}{c}}
  b&{c + a} \\ 
  c&c 
\end{array}} \right|$

$ = 2c\left( {ab + {b^2} – bc} \right) – 2b\left( {bc – {c^2} – ac} \right)$

$ = 2abc + 2c{b^2} – 2b{c^2} – 2{b^2}c + 2b{c^2} + 2abc$

$ = 4abc$