What is the value of a b c a - b b - c c - a b + c c + a a + b = ?

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

$(R_2\to R_2-R_1)$

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

$(R_3\to R_3+R_2)$

And taking out $-1$ As common

$-1\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

Solving the determinant we get

$=-1\left[a\right(cb-a^2)-b(b^2-ac)+c(ab-c^2\left)\right]$

$=-1[3abc-a^3-b^3-c^3]$

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