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${(x + 5 y - 3 z)}^{3} + {(x - 5 y + 3 z)}^{3} + 6 x (x + 5 y - 3 z) (x - 5 y + 3 z) = 8 x^{3}$

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Solution

$(x + 5 y - 3 z)^{3} + (x - 5 y + 3 z)^{3} + 6 x (x + 5 y - 3 z) (x - 5 y + 3 z)$ it can be written as
$(x + 5 y - 3 z)^{3} + (x - 5 y + 3 z)^{3} + 3 (x + 5 y - 3 z) (x - 5 y + 3 z) ((x + 5 y - 3 z) + (x - 5 y + 3))$ which is in the form of $(a + b)^{3} = a^{3} + b^{3} + 3 a b (a + b) = (a + b)^{3}$
$⟹ (x + 5 y - 3 z + x - 5 y + 3 z)^{3} = (2 x)^{3} = 8 x^{3}$

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