<s>[INST] Solving multivariable 4 grade equation Find the integer solutions of the following equation:
$$8x^4+y^4=6x^2y^2$$
I tried to reconfigure the equation using the notable identity $(a-b)^2=a^2+b^2-2ab$, but due to the numeric coeficients I was not able
 [/INST] You can factor
$8x^4-6x^2y^2+y^4$ as
$9x^4-6x^2y^2+y^4-x^4=(3x^2-y^2)^2-(x^2)^2=(4x^2-y^2)(2x^2-y^2)$
So the equation becomes
$(4x^2-y^2)(2x^2-y^2)=0$
At least one of these two factors has to be equal to 0
But $2x^2=y^2$ is impossible, because doubling a square changes the exponent of $2$ present in the prime factorization of the square from even to odd, which makes it a non-square integer. So:
$4x^2=y^2$
$y=\pm 2x$
The solutions are in the form $(k;\pm2k)$
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