Question 461590
Not both of a and b are equal to zero, by the given.  Assume one of them, say a, is zero, but {{{b <> 0}}}.  Then {{{b^4 > 0*b*(0+b) = 0}}}, which is always true.
Hence assume that both a and b are positive.

Then 

{{{(a-b)^2(a^2 + ab + b^2) > 0}}}
<==> {{{(a-b)(a-b)(a^2 + ab + b^2) > 0}}}
<==> {{{(a^3-b^3)(a - b) > 0}}}
<==> {{{a^3(a-b) - b^3(a-b) > 0}}}
<==> {{{a^3(a-b) + b^3(b-a) > 0 }}}

<==> {{{a^4 - a^3b + b^4 - ab^3 > 0}}}

<==> {{{a^4 + b^4 > a^3b + ab^3}}}

<==> {{{a^4 + b^4 > ab(a^2 + b^2)}}}