The value of  b + c a a b c + a b c c a + b  is 

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

Applying $R_1\to R_1+R_2+R_3$

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

$\Rightarrow △=\left|\begin{array}{ccc}2\left(b+c\right)& 2\left(a+c\right)& 2\left(a+b\right)\\ b& c+a& b\\ c& c& a+b\end{array}\right|$

$\Rightarrow △=2\left|\begin{array}{ccc}b+c& a+c& a+b\\ b& c+a& b\\ c& c& a+b\end{array}\right|$

Applying $R_1\to R_1-R_2$

$\Rightarrow △=2\left|\begin{array}{ccc}b+c-b& a+c-(c+a)& a+b-b\\ b& c+a& b\\ c& c& a+b\end{array}\right|$

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

Applying $R_3\to R_3-R_1$

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

Applying $R_2\to R_2-R_3$

$\Rightarrow △=2\left|\begin{array}{ccc}c& 0& a\\ b& a& 0\\ 0& c& b\end{array}\right|$

$\Rightarrow △=2\left[c\left(ab-0\right)-0+a\left(bc-0\right)\right]$

$\Rightarrow △=2\left[abc+abc\right]$

$\Rightarrow △=4abc.$