Let $w = \frac{1+\sqrt{-11}}{2}$ and let $R = \mathbb{Z}[w] = \{a + b w \mid a, b \in \mathbb{Z}\}$. Show that $R$ is a Euclidean domain.

Question

Let $w = \frac{1+\sqrt{-11}}{2}$ and let $R = \mathbb{Z}[w] = \{a + b w \mid a, b \in \mathbb{Z}\}$. Show that $R$ is a Euclidean domain.

I used the norm function $\phi(a+bw)=a^2 + ab + 3b^2$ as my Euclidean function. I have shown that $\phi(a+bw)\leq\phi((a+bw)(c+dw))$. Now I am stuck at showing $$\exists q, r \in R \; \text{such that} \; a + bw = (c + dw)q + r \; \text{and} \; (r = 0 \; \text{or} \; \varphi(r) < \varphi(c + dw))$$ I got myself into lots of "hairy" calculations without any result.

Any help is appreciated.