<s>[INST] Solving $y^2(x^2+1) +x^2(y^2+16) =448$ $y^2(x^2+1) +x^2(y^2+16) =448$
The task is to find all solutions in integers $(x,y)$. This is the fourth question of rmo 1st stage.The solution here is not complete.  I have tried to solve unable to.
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 [/INST] First, this equation is symmetric under $x\to-x$ and $y\to-y$, independently. Therefore we can restrict attention to $x,y>0$ and such a solution will represent four actual solutions (we easily see that $x=0$ or $y=0$ are not possible integer solutions).
Solve with the quadratic formula to find
$$y=4\frac{\sqrt{28-x^2}}{\sqrt{1+2x^2}}.$$
The allowed integer values of $x$ are $1$, $2$, $3$, $4$, $5$. Check these by hand: $x=1$ and $x=3$ give integer values of $y$, so there are eight solutions total:
$$(\pm 1,\pm 12)$$
$$(\pm 3,\pm 4)$$
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