$4+\sqrt{10}$ is not prime in $R$.

Question

I want to prove following:

Let $R$ be the subring $\{a+b\sqrt{10}\mid a,b\in \mathbb{Z}\}$ of $\mathbb{R}$. Then $4+\sqrt{10}$ is not a prime in $R$.

To show $4+\sqrt{10}$ is not a prime, I need to come up with an example such that $4+\sqrt{10}\mid ab$ but $4+\sqrt{10}\not\mid a$ and $4+\sqrt{10}\not\mid b$. However, I wasn't able to come up with an example. Can anyone help me? Thanks.

Answer 1

$(4+\sqrt{10})(4-\sqrt{10})=16-10=6=2\cdot3$, but neither $2$ nor $3$ can be written in the form $(4+\sqrt{10})(a+b\sqrt{10})$.