find all the divisors of $6$ and $4+2\sqrt{5}$,then find $\gcd(6,4+2\sqrt{5})$

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By inspection we see that the divisors of $6$ are $1,2,3,6$

For $4+2\sqrt{5}$ we have $4+2\sqrt{5}=2(2+\sqrt{5})$

showing that $\gcd(6,4+2\sqrt{5})=2$

Is this method correct; if not, how can I do this successfully?