Prove: `|a^3 2a b^3 2b c^3 2c|=2(a-b)(b-c)(c-a)(a+b+c)`

Explanation

We have to prove `|(a^3, 2,a),( b^3, 2,b),( c^3, 2,c)|=2(a-b)(b-c)(c-a)(a+b+c)`

Then let

`Delta =|(a^3, 2,a),( b^3, 2,b),( c^3, 2,c)|`

Now apply `R_1->R_1-R_2, R_2->R_2-R_3`

`implies Delta =|(a^3-b^3, 0,a-b),( b^3-c^3, 0,b-c),( c^3, 2,c)|`

`implies Delta =|((a-b)(a^2+b^2+ab), 0,a-b),((b-c)( b^2+c^2+bc), 0,b-c),( c^3, 2,c)|`

`implies Delta=(a-b)(b-c)|(a^2+b^2+ab, 0,1),( b^2+c^2+bc, 0,1),( c^3, 2,c)|`

Now apply `R_1->R_1-R_2`

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