If $x = \sqrt{2} + \sqrt{3} + \sqrt{6}$ is a root of $x^4 + ax^3 + bx^2 + cx + d$, where $a, b, c$ and $d$ are integers, what is $|a + b + c + d|$? From the Pre-Regional Mathematics Olympiad, 2019:

If $x = \sqrt{2} + \sqrt{3} + \sqrt{6}$ is a root of $x^4 + ax^3 + bx^2 + cx + d$, where $a, b, c$ and $d$ are integers, what is $|a + b + c + d|$?

I have provided one answer below, and would be interested in alternative solutions.

The only part missing in the existing answer is the fact that the minimal polynomial of $a=\sqrt{2}+\sqrt{3}+\sqrt{6}$ has degree $4$, i.e. the degree of the extension $|\Bbb{Q}[a]:\Bbb{Q}|$ is $4$. Since $a$ is a root of an integer  polynomial of degree $4$, the degree of the extension is $1, 2$ or $4$. It cannot be $1$ since $a$ is not rational. If the degree is $2$ then for some rational $p,q$ you have $a^2+pa+q=0$ or $11+2(\sqrt{6}+2\sqrt{3}+3\sqrt{2})+pa+q=0$. But that is impossible. So, indeed, the degree is $4$.
Another way to prove it is to consider homomorphisms from $\Bbb{Q}[a]$ to $\Bbb{Q}[\sqrt{2},\sqrt{3}]$ (which has degree $4$). There are $4$ of those.
