Pt$\Leftrightarrow x^{2}-y^{2}+2x+4y+1=2\sqrt{(x^{2}+2x+3)(-y^{2}+4y-2)}$$\Leftrightarrow x^{2}+2x+3-2\sqrt{(x^{2}+2x+3)(-y^{2}+4y-2)}+(-y^{2}+4y-2)=0$
$\Leftrightarrow (\sqrt{x^{2}+2x+3}-\sqrt{-y^{2}+4y-2})^2=0$
$\Leftrightarrow \sqrt{x^{2}+2x+3}=\sqrt{-y^{2}+4y-2}$
$\Leftrightarrow x^{2}+2x+3=-y^{2}+4y-2$
$\Leftrightarrow (x+1)^{2}+(y-2)^{2}=0$
$\Leftrightarrow \begin{cases}x=-1 \\ y=2 \end{cases}$
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