Ta có: $a + b + c = 0$

$\Rightarrow (a + b + c)^2 = 0$

$\Rightarrow a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = 0$

$\Rightarrow a^2 + b^2 + c^2 = -2(ab + bc + ca)$

$\Rightarrow (a^2 + b^2 + c^2)^2 = [-2(ab + bc + ca)]^2$

$\Rightarrow a^4 + b^4 + c^4 + 2a^2b^2 + 2b^2c^2 + 2c^2a^2 = 4(a^2b^2 + b^2c^2 + c^2a^2 + 2ab^2c + 2a^2bc + 2abc^2)$

$\Rightarrow a^4 + b^4 + c^4 = 4a^2b^2 + 4b^2c^2 + 4c^2a^2 + 8a^2bc + 8ab^2c + 8abc^2 - 2a^2b^2 - 2b^2c^2 - 2a^2c^2$

$\Rightarrow a^4 + b^4 + c^4 = 2a^2b^2 + 2b^2c^2 + 2c^2a^2 + 8abc(a + b + c)$

$\Rightarrow a^4 + b^4 + c^4 = 2a^2b^2 + 2b^2c^2 + c^2a^2$

$\Rightarrow a^4 + b^4 + c^4 = 2a^2b^2 + 2b^2c^2 + 2c^2a^2 + 2abc(a + b + c) \quad (\text{Vì } a + b + c = 0)$

$\Rightarrow a^4 + b^4 + c^4 = 2a^2b^2 + 2b^2c^2 + 2c^2a^2 + 2a^2bc + 2ab^2c + 2abc^2$

$\Rightarrow a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2 + a^2bc + ab^2c + abc^2)$

$\Rightarrow a^4 + b^4 + c^4 = 2(ab + bc + ca)^2 \quad (\text{đpcm})$

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