Need to find the non negative integral solutions for the equation
I have tried various methods for simplifying the RHS and LHS but could not arrive at the solution, so any help will be appreciated
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First get a bound for possible solutions. Suppose without loss of generality that $x \leqslant y$. Then you have
$$(y+1)^2 = y^2 + y + y + 1 > xy + y + x = y^3 + x^3 \geqslant y^3.$$
That gives you $y \leqslant 2$. But for $y = 2$, we have $(2+1)^2 = 9$ and $2^3 = 8$, so the equality could only hold if $x = y$, or the first inequality would lose too much, and $x = 0$, or the second inequality would lose too much. So actually $y \leqslant 1$. Then it is a simple exhaustive search that yields the solutions $(0,0),\, (0,1),\, (1,0)$.