Factorize:

We have:

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

We have to split $3 \sqrt{3}$ into two numbers such that their sum is $3 \sqrt{3}$ and their product is 6 , i.e., $2 \times 3$.

Clearly, $2 \sqrt{3}+\sqrt{3}=3 \sqrt{3}$ and $2 \sqrt{3} \times \sqrt{3}=6$.

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

$=2 x(x+\sqrt{3})+\sqrt{3}(x+\sqrt{3})$

$=(x+\sqrt{3})(2 x+\sqrt{3})$