Ta có: (1) \(\Leftrightarrow (\sqrt{a}+\sqrt{a+2})^{2}\leq 4(a+1)\)
\(\Leftrightarrow a+a+2+2\sqrt{a(a+2)}\leq 4(a+1)\)
\(\Leftrightarrow 2\sqrt{a(a+2)}\leq 2(a+1) \\
\Leftrightarrow \sqrt{a(a+2)}\leq a+1\)
\(\Leftrightarrow a(a+2)\leq (a+1)^{2} \\
\Leftrightarrow 2a+a^{2}\leq a^{2}+2a+1\)
\(\Leftrightarrow 0\leq 1\) đúng
\(\Rightarrow \)đpcm.
Dấu bằng không xảy ra.