$Đk:x\ge0;x\ne1;x\ne4$
$P=\bigg(\dfrac{4\sqrt{x}}{\sqrt{x}+2}-\dfrac{8x}{x-4}\bigg):\bigg(\dfrac{\sqrt{x}+2}{\sqrt{x}-2}+3\bigg)\\ =\dfrac{4\sqrt{x}(\sqrt{x}-2)-8x}{(\sqrt{x}-2)(\sqrt{x}+2)}.\dfrac{\sqrt{x}-2}{3(\sqrt{x}-2)+\sqrt{x}+2}\\ =\dfrac{4x-8\sqrt{x}-8x}{(\sqrt{x}+2)4(\sqrt{x}-1)}\\ =\dfrac{-4\sqrt{x}(\sqrt{x}+2)}{(\sqrt{x}+2)4(\sqrt{x}-1)}\\ =\dfrac{-\sqrt{x}}{\sqrt{x}-1}$
$P=-4\Leftrightarrow \dfrac{-\sqrt{x}}{\sqrt{x}-1}=-4\\ \Leftrightarrow \sqrt{x}=4\sqrt{x}-4\\ \Leftrightarrow 3\sqrt{x}=4\\ \Leftrightarrow \sqrt{x}=\dfrac{4}{3}\\ \Leftrightarrow x=\dfrac{16}{9}\ (t/m)$
Vậy với $x=\dfrac{16}{9}$ thì $P=-4$
