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.

We can rewrite $x =\sqrt{2} + \sqrt{3} + \sqrt6$ as
$$\begin{aligned}(x-\sqrt2)^2 &= (\sqrt2 + \sqrt6)^2\\
x^2 - 2\sqrt2x+2 &= 9 + 6\sqrt2\\
x^2-7&=2\sqrt2\space(x+3)\end{aligned}$$
On squaring both sides of the equation,
$$x^4-22x^2-48x-23=0$$
Therefore, $a = 0, b = -22, c = -48, d = -23$, implying $|a+b+c+d| = 93$
