Factorize:

We have:

$\sqrt{2} x^{2}+3 x+\sqrt{2}$

We have to split 3 into two numbers such that their sum is 3 and their product is 2 , i.e., $\sqrt{2} \times \sqrt{2}$.

Clearly, $2+1=3$ and $2 \times 1=2$

$\therefore \sqrt{2} x^{2}+3 x+\sqrt{2}=\sqrt{2} x^{2}+2 x+x+\sqrt{2}$

$=\sqrt{2} x(x+\sqrt{2})+1(x+\sqrt{2})$

$=(x+\sqrt{2})(\sqrt{2} x+1)$