Which of these two factorizations of $5$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is more valid? $$5 = (-1) \left( \frac{3 - \sqrt{29}}{2} \right) \left( \frac{3 + \sqrt{29}}{2} \right)$$
or 
$$5 = \left( \frac{7 - \sqrt{29}}{2} \right) \left( \frac{7 + \sqrt{29}}{2} \right)?$$
$\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is supposed to be a unique factorization domain, so the two above factorizations are not distinct. But I can divide factors in both of them by units to obtain yet more seemingly different factorizations.
The presence of the unit $-1$ in the first factorization does not trouble me, since for example on this page http://userpages.umbc.edu/~rcampbel/Math413Spr05/Notes/QuadNumbFlds.html the factorization of $3$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{13})}$ is given as $(-1)(7 - 2 \sqrt{13})(7 + 2 \sqrt{13})$.
I honestly find rings of complex numbers far easier to understand!

Note that
$$ \frac{7-\sqrt{29}}2 \cdot \frac{5+\sqrt{29}}2 = \frac{3+\sqrt{29}}2 $$
and
$$ \frac{5+\sqrt{29}}2 \cdot \frac{-5+\sqrt{29}}2 = 1 $$
So the two factorizations are equally good, just related by unit factors. (Remember that even in an UFD, factorizations are only unique up to associatedness).
