$\Large x^{4} = 4x^{3}y + 18x^{2}y^{2} + 28 x y^{3} +15 y^{4}$

Find the number of positive integral solutions to the above equation.

The answer is 0.

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Factorizing the polynomial we get

$(x+y)(15{ y }^{ 3 }+13x{ y }^{ 2 }+5{ x }^{ 2 }y-{ x }^{ 3 })=0$

$\Rightarrow \quad x+y=0\quad \quad OR\quad \quad x+y=0.1404239...(irrational)$

RHS cannot be true as x and y are integers.

LHS cannot be true as x and y are positive.