Factorize:

We have:

$\sqrt{5} x^{2}+2 x-3 \sqrt{5}$

We have to split 2 into two numbers such that their sum is 2 and product is $(-15)$, i.e., $\sqrt{5} \times(-3 \sqrt{5})$.

Clearly, $5+(-3)=2$ and $5 \times(-3)=-15$.

$\therefore \sqrt{5} x^{2}+2 x-3 \sqrt{5}=\sqrt{5} x^{2}+5 x-3 x-3 \sqrt{5}$

$=\sqrt{5} x(x+\sqrt{5})-3(x+\sqrt{5})$

$=(x+\sqrt{5})(\sqrt{5} x-3)$