Factorize:

We have:

$7 x^{2}+2 \sqrt{14} x+2$

We have to split $2 \sqrt{14}$ into two numbers such that their sum is $2 \sqrt{14}$ and product is 14 .

Clearly, $\sqrt{14}+\sqrt{14}=2 \sqrt{14}$ and $\sqrt{14} \times \sqrt{14}=14$

$\therefore 7 x^{2}+2 \sqrt{14} x+2=7 x^{2}+\sqrt{14} x+\sqrt{14} x+2$

$=\sqrt{7} x(\sqrt{7} x+\sqrt{2})+\sqrt{2}(\sqrt{7} x+\sqrt{2})$

$=(\sqrt{7} x+\sqrt{2})(\sqrt{7} x+\sqrt{2})$

$=(\sqrt{7} x+\sqrt{2})^{2}$