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The question is not well formed. you have forgotten the - 3 x y z term on the LHS.

(x + y + z) * [ (x - y)² + (y - z)² + (z - x)² ]

= (x + y + z) * [ x² + y² - 2 x y + y² + z² - 2 y z + z² + x² - 2 z x ]

= (x + y + z) * 2 * [ x² + y² + z² - x y - y z - z x ]

= 2 [ x³ + x y² + x z² - x² y - x y z - z x² + x² y + y³ + y z² - x y² - y² z - x y z

+ x² z + y² z + z³ - x y z - y z² - x z² ]

= 2 [ x³ + y³ + z³ - 3 x y z] + 2 [ 0 ]

Hence you get the answer.

(x + y + z) * [ (x - y)² + (y - z)² + (z - x)² ]

= (x + y + z) * [ x² + y² - 2 x y + y² + z² - 2 y z + z² + x² - 2 z x ]

= (x + y + z) * 2 * [ x² + y² + z² - x y - y z - z x ]

= 2 [ x³ + x y² + x z² - x² y - x y z - z x² + x² y + y³ + y z² - x y² - y² z - x y z

+ x² z + y² z + z³ - x y z - y z² - x z² ]

= 2 [ x³ + y³ + z³ - 3 x y z] + 2 [ 0 ]

Hence you get the answer.