Show that `|[a, b-c, c+b] , [a+c, b, c-a] , [a-b, b+a, c]|=(a+b+c)(a^2+b^2+c^2)`

Explanation

Let `Delta= |{:(a,,b-c,,c+b),(a+c,,b,,c-a),(a-c,,b+a,,c):}|`
`=(1)/(a) |{:(a^(2),,b-c,,c+b),(a^(2)+ac,,b,,c-a),(a^(2)-ab,,b+a,,c):}|`
[Multiplying first bya]
`=(1)/(a) |{:(a^(2)+b^(2)+c^(2),,b-c,,c+b),(a^(2)+b^(2)+c^(2),,b,,c-a),(a^(2)+b^(2)+c^(2),,b+a,,c):}|`
[Applying `C_(1)to C_(1) +bC_(2)+cC_(3)`]
`=(1)/(a)(a^(2)+b^(2)+c^(2)) |{:(1,,b-c,,c+b),(1,,b,,c-a),(1,,b+a,,c):}|`
[Taking `a^(2)+b^(2)+c^(2)` common from `C_(1)`]
`=(1)/(a)(a^(2)+b^(2)+c^(2)) |{:(1,,b-c,,c+b),(1,,c,,-a-b),(0,,a+c,,-b):}|`
[Applying `R_(2) to R_(2) -R_(1)" and "R_(3) to R_(3)-R_(1)`]
`=(1)/(a)(a^(2)+b^(2)+c^(2))xx1xx |{:(c,,-a-b),(a_c,,-b):}|`
[Expanding along `C_(1)`]
=`(1)/(a)(a^(2) +b^(2)+c^(2))(-bc+a^(2)+ac+ba+bc)`
`=(a^(2)+b^(2)+c^(2))(a+b+c)`