<s>[INST] $(5 + \sqrt{2})(2-\sqrt{2})=(11-7\sqrt{2})(2+\sqrt{2})$ - Unique factorization? Personal question : We know that $5 + \sqrt{2}$, $2-\sqrt{2}$, $11-7\sqrt{2}$ and $2+\sqrt{2}$ are irreductible in $\mathbb{Z}[\sqrt{2}]$ and that $$(5 + \sqrt{2})(2-\sqrt{2})=(11-7\sqrt{2})(2+\sqrt{2}).$$ Why this fact doesn't contradict the unique factorization in $\mathbb{Z}[\sqrt{2}]$?
Is it because $(5 + \sqrt{2})(2-\sqrt{2})=(5 + \sqrt{2})(2+\sqrt{2})(3-2\sqrt{2})=(11 -7\sqrt{2})(2+\sqrt{2})$?
Is anyone could give me a full explication in ''Answer the question''? 
 [/INST] Unique factorization only means unique factorization upto units. 
The units in ${\mathbb Z}[\sqrt{2}]$ are the elements of the form $(1 + \sqrt{2})^n$ with $n \in {\mathbb Z}$.
For your two factorizations:
$$5 + \sqrt{2} = (3 + 2\sqrt{2}) (11 - 7 \sqrt{2}) = (1 + \sqrt{2})^2 (11 - 7 \sqrt{2})$$
so they differ by a unit.
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