$\Leftrightarrow \sqrt{x+3}-\sqrt[4]{x}-1+\sqrt{x}-2\sqrt[4]{x}+1\geq 0$
$\Leftrightarrow \frac{x+3-(\sqrt[4]{x}+1)^2}{\sqrt{x+3}+\sqrt{x}+1}+(\sqrt[4]{x}-1)^2\geq 0$
$\Leftrightarrow \frac{x+2-\sqrt{x}-2\sqrt[4]{x}}{\sqrt{x+3}+\sqrt[4]{x}+1}+(\sqrt[4]{x}-1)^2\geq 0$
$\Leftrightarrow \frac{\sqrt{x}(\sqrt{x}-1)-2(\sqrt[4]{x}-1)}{M_1}+(\sqrt[4]{x}-1)^2\geq 0$
$\Leftrightarrow \frac{(\sqrt[4]{x}-1)(\sqrt[4]{x^3}+\sqrt[4]{x^2}-2)}{M_1}+(\sqrt[4]{x}-1)^2\geq 0$
$\Leftrightarrow \frac{(\sqrt[4]{x}-1)^2(\sqrt{x}+2)}{M_1}+(\sqrt[4]{x}-1)^2\geq 0$
$\Leftrightarrow (\sqrt[4]{x}-1)^2.(\frac{\sqrt{x}+2}{M_1}+1)\geq 0$ ~> lđ
$\rightarrow $ đpcm~~