In the ring of $\mathbb Z[i]$ use our division algorithm divide $7+2i$ by $2+i$.
$\frac{7+2i}{2+i}\cdot \frac{2-i}{2-i}=\frac{16-3i}{5}=\frac{16}{5}-\frac{3}{5}i$
Does this mean you cannot divide $7+2i$ by $2+i$, because the real and imaginary parts are $\frac{16}{5}$ and $\frac{3}{5}i$ which are not integers.