I could use some help with this.
I know that $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5\} }|a,b\in\mathbb{Z}\}$. I then put $$0=(a+b\sqrt{-5})(c+d\sqrt{-5})=ac-5bd+(ad+bc)\sqrt{-5}$$ which leaves me with $$0=ac-5bd$$ and $$0=(ad+bc)\sqrt{-5}$$
now how do I proceed from here? I know that I have to prove $a+b\sqrt{-5}=0$ for $c+d\sqrt{-5}\neq0$.
For the first part : numbers in $\mathbb{Z}[\sqrt{-5}]$ are in particular complex numbers. What happens when a product of two complex numbers is zero ?
For the second part : to show that your elements irreducible, try to see that no element has its norm equal to $2$ or $3$ (the norm of $a+b\sqrt{-5}$ is $a^2+5b^2$).