If $4xy+3=c^2+3d^2$, is $xy$ necessarily a square?

Question

I have a polynomial which, simplified, ends up in the form $$4xy+3 = c^2+3d^2.$$

Evidently $4xy+3$ is of the form $a^2+3b^2$, in light of the equality. But does $$ c^2 + 3d^2 = 4xy + 3 = xy(2)^2 + 3(1)^2 $$ necessarily force $xy$ to be a square? I just can't see how to prove it.

Thanks.

Answer 1

This answer is just to remove this post from the Unanswered queue. As per the counterexample given by @Tomas, the answer is negative: $xy$ does not have to be a square.

Answer 2

For the equation: $4xy+3=c^2+3d^2$

Can be written like such a simple solution:

$x=2a^2+6t^2-6at-2a+3t-1$

$y=2a^2+6t^2-6at-a+3t-1$

$c=2a^2+6t^2-6at-2$

$d=2a^2+6t^2-6at-2a+4t-1$

$a,t$ - what some integers any sign.