Finding the ring of integers of $\mathbb Q[\alpha]$ with $\alpha^5=2\alpha+2$.

Let's look at the first part. Let $R$ be the ring of integers of $\mathbb Q[\alpha]$. Let $\mathfrak p$ be a maximal ideal of $R$ containing $2$. Then $\alpha^5=2(1+\alpha)\in \mathfrak p$, hence $\alpha\in \mathfrak p$. Moreover $1+\alpha\notin \mathfrak p$, so $2\in \mathfrak p^5R_\mathfrak p$. This implies that the ramification index at $\mathfrak p$ is at least $5$. But $\mathbb Q[\alpha]$ has degree $5$, so the ramification index is exactly $5$, the residue extension is trivial and there is only one prime above $2$. By Problem 21 just before the one you are considering, $2^4$ divides the discriminant of $R$.

Now the discriminant of $X^5-2X-2$ is $2^4.3.13.67$. So it is equal to the discriminant of $R$ and therefore $\mathbb Z[\alpha]=R$.

The seconde part should be solved similarly.

Edit To see the ramification index at $\mathfrak p$ is at least $5$ without using localization (I don't know how things are organized in Marcus), we can just compare ideal decompositions of $\alpha^5 R$ and $2(1+\alpha)R$, noticing that no prime ideal can contains at the same time $\alpha$ and $1+\alpha$.