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Prove: |[b+c,a,a],[b,c+a,b],[c,c,a+b]|=...

Prove: `|[b+c,a,a],[b,c+a,b],[c,c,a+b]|=4a b c`

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`L.H.S. = |[b+c,a,a],[b,c+a,b],[c,c,a+b]|`
Applying `R_1->R_1-R_2-R_3`
`= |[0,-2c,-2b],[b,c+a,b],[c,c,a+b]|`
`=-2|[0,c,b],[b,c+a,b],[c,c,a+b]|`
`=-2[-c(ab+b^2-bc)+b(bc-c^2-ca)]`
`=-2[-abc-cb^2+bc^2+b^2c-bc^2-abc]`
`=-2(-2abc) = 4abc = R.H.S.`
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