The polynomial is $x^4-2x^3-2x+4$ which is reducible over $\mathbb Q$ as it has a root $x=2$. The cubic resolvent for this polynominal is $x^3-16x-20$. It is irreducible and its determinant equals $2^4\cdot 349$, meaning that the Galois group for the resolvent is $S_3$. I found information on how to determine the Galois group for irreducible quartic polynomials based on the information about its cubic resolvent, but nothing about the case when it's reducible.
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5That's a quartic, not a quadratic. After factoring, it is $(x-2)(x^3-2)$ so you really only care about $x^3-2$, which is a standard example – lulu Dec 27 '21 at 14:07
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@lulu Thank you. I am sorry, English is not my first language. So will the Galois group of $x^3-2$ be equal to the one of quartic polynominal? – ohstapit Dec 27 '21 at 14:20
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Yes, as they have the same splitting field. – lulu Dec 27 '21 at 14:20