1^st way : We have

\(\sqrt{8}^2=8=6+2\)

\(\left(\sqrt{5}+1\right)^2=5+2\sqrt{5}+1=6+2\sqrt{5}\)

Cause \(2\sqrt{5}\ge2\sqrt{4}=4>2\)

So \(\sqrt{8}< \sqrt{5}+1\)

2^st way :

\(\sqrt{8}< \sqrt{9}=3=2+1=\sqrt{4}+1< \sqrt{5}+1\)

So ...... 

1st way: \(\sqrt{8}=\text{2.82842712475}\)

               \(\sqrt{5}+1=\text{3.2360679775}\)

So \(\sqrt{8}< \sqrt{5}+1\)

2nd way:

\(\sqrt{8}< \sqrt{9}=3\)

\(\sqrt{5}+1>\sqrt{4}+1=2+1=3\)

So \(\sqrt{8}< \sqrt{5}+1\)