a(b³ - c³) + b(c³ - a³) + c(a³ - b³)

= a(b³ - c³ ) + b( c³ - b³ + b³ - a³) + c(a³ - b³)

= a(b³ - c³) + b(c³ - b³) + b(b³ - a³) + c(a³ - b³)

=[a(b3−c3)−b(b3−c3)]−[b(a3−b3)−c(a3−b3)]=[a(b3−c3)−b(b3−c3)]−[b(a3−b3)−c(a3−b3)]

= (b³ - c³)(a - b) - (a³- b³)(b - c)

= (b - c)(b² + bc + c²)(a - b) - (a - b)(a² + ab + b²)(b - c)

= (b - c)(a - b)(b² + bc + c² - a² + ab - b²)

= (b - c)(a - b) [ (c²  - a²) + (bc - ab) ]

= (b - c)(a - b) [ (c - a)(c + a) + b(c - a) ]

= (b - c)(a -b) [ (c - a)(c + a + b) ]

= (a- b)(b - c)(c - a)(a + b + c)

a + b + c = 0

=> (a + b + c)² = 0

=> a² + b² + c² + 2ab + 2bc + 2ca = 0

=> a² + b² + c² = -2(ab + 2bc + 2ca)

=> (a² + b² + c²)² = [-2(ab + bc + ca)]²

=> a⁴ + b⁴ + c⁴ + 2a²b² + 2b²c² + 2c²a² = 4(a²b² + b²c² + c²a² + 2ab²c + 2a²bc + 2abc²) 

=> a⁴ + b⁴ + c⁴ = 4a²b² + 4b²c² + 4c²a² + 8a²bc + 8ab²c + 8abc² - 2a²b² - 2b²c² - 2a²c²

=> a⁴ + b⁴ + c⁴ = 2a²b² + 2b²c² + 2c²a² + 8abc(a + b + c)

=> a⁴ + b⁴ + c⁴= 2a²b² + 2b²c² + c²a²

=> a⁴ + b⁴ + c⁴ = 2a²b² + 2b²c² + 2c²a² + 2abc(a + b + c) (Vì a + b + c = 0)

=> a⁴ + b⁴ + c⁴ = 2a²b² + 2b²c² + 2c²a² + 2a²bc + 2ab²c + 2abc²

=>  a⁴ + b⁴ + c⁴ = 2(a²b² + b²c² + c²a² + a²bc + ab²c + abc²) 

=> a⁴ + b⁴ + c⁴ = 2(ab + bc + ca)² (đpcm)