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Simplify: ((a^2-b^2)^3+(b^2-c^2)^3+(c^2-...

Simplify: `((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)^3)`

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Given:`((a^2−b^2)^3+(b^2−c^2)^3+(c^2−a^2)^3)/((a−b)^3+(b−c)^3+(c−a)^3)`
We know that if `x+y+z=0`then `x^3+y^3+z^3=3xyz`
Since,
`=>a^2−b^2+b^2−c^2+c^2−a^2=0`
therefore,
`(a^2−b^2)^3+(b^2−c^2)^3+(c^2−a^2)^3=3(a^2−b^2)(b^2−c^2)(c^2−a^2)`
Also,
`=>a−b+b−c+c−a=0`
`(a−b)^3+(b−c)^3+(c−a)^3=3(a−b)(b−c)(c−a)`
Therefore,
`((a^2−b^2)^3+(b^2−c^2)^3+(c^2−a^2)^3)/((a−b)^3+(b−c)^3+(c−a)^3)`
`=(3(a^2−b^2)(b^2−c^2)(c^2−a^2))/(3(a−b)(b−c)(c−a))`
`=((a−b)(a+b)(b−c)(b+c)(c−a)(c+a))/((a−b)(b−c)(c−a))`
`=(a+b)(b+c)(c+a)`
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