Prove that `Delta = |{:(1,,1,,1),(a,,b,,c),(bc+a^(2),,ac+b^(2),,ab+c^(2)):}|`
`= 2(a-b)(b-c)(c-a)`

Explanation

`Delta = |{:(1,,1,,1),(a,,b,,c),(bc+a^(2),,ac+b^(2),,ab+c^(2)):}|`
` |{:(1,,1,,1),(a,,b,,c),(bc,,ac,,ab):}|+|{:(1,,1,,1),(a,,b,,c),(a^(2),,b^(2),,b^(2)):}|`
`" Now " |{:(1,,1,,1),(a,,b,,c),(bc,,ac,,ab):}|`
`=(1)/(abc) |{:(a,,b,,c),(a^(2),,b^(2),,c^(2)),(abc,,abc,,abc):}|" "underset("a,b,c respectively]")("[Multiplying "C_(1),C_(2),C_(3)" by")`
`=|{:(a,,b,,c),(a^(2),,b^(2),,c^(2)),(1,,1,,1):}|`
`=|{:(1,,1,,1),(a,,b,,c),(a^(2),,b^(2),,c^(2)):}|`
`rArr Delta =2 |{:(1,,1,,1),(a,,b,,c),(a^(2),,b^(2),,c^(2)):}| = 2(a-b)(b-c)(c-a)`