LHS = \(\begin{vmatrix}
b + c & a + b & a \\[0.3em]
c + a & b + c & b \\[0.3em]
a + b &c + a &c
\end{vmatrix}\)
= (a + b + c)[0 – 0 + 1{(a – c)(b – c) – (a – b) (b – a)}]
= (a + b + c){(ab – ca – bc + c2) – (ab – a2 – b2 + ab)}
= (a + b + c)(ab – ca – bc + c2 – ab + a2 + b2 – ab)
= (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
= a3 + b3 + c3 – 3abc – R.H.S. Proved