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(B) Let $\Delta = \left| {\begin{array}{ccc} b + c & a - b & a \ c + a & b - c & b \ a + b & c - a & c \end{array}} ight|$.Applying the row operat

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$3abc - a^3 - b^3 - c^3$

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(B) Let $\Delta = \left| {\begin{array}{ccc} b + c & a - b & a \ c + a & b - c & b \ a + b & c - a & c \end{array}} ight|$.Applying the row operation $R_1 	o R_1 + R_2 + R_3$:$\Delta = \left| {\begin{array}{ccc} 2(a + b + c) & 0 & a + b + c \ c + a & b - c & b \ a + b & c - a & c \end{array}} ight|$.Taking $(a + b + c)$ common from $R_1$:$\Delta = (a + b + c) \left| {\begin{array}{ccc} 2 & 0 & 1 \ c + a & b - c & b \ a + b & c - a & c \end{array}} ight|$.Expanding along $R_1$:$\Delta = (a + b + c) [2(c(b - c) - b(c - a)) - 0 + 1((c + a)(c - a) - (a + b)(b - c))]$.$\Delta = (a + b + c) [2(bc - c^2 - bc + ab) + (c^2 - a^2 - (ab - ac + b^2 - bc))]$.$\Delta = (a + b + c) [2(ab - c^2) + c^2 - a^2 - ab + ac - b^2 + bc]$.$\Delta = (a + b + c) [ab - a^2 - b^2 - c^2 + ac + bc]$.$\Delta = -(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$.Using the identity $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$:$\Delta = -(a^3 + b^3 + c^3 - 3abc) = 3abc - a^3 - b^3 - c^3$.

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

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