<s>[INST] Given $x^2 + 4x + 6$ as factor of $x^4 + ax^2 + b$, then $a + b$ is I got this task two days ago, quite difficult for me, since I have not done applications of Vieta's formulas and Bezout's Theorem for a while. If can someone solve this and add exactly how I am supposed to use these two theorem's on this task, I would be thankful.

Given $x^2 + 4x + 6$ as factor of $x^4 + ax^2 + b$, then $a + b$ is equal to?

 [/INST] \begin{align*}
x^4+ax^2+b & = x^2\color{blue}{(x^2+4x+6)}-4x\color{blue}{(x^2+4x+6)}+\\
&(a+10)\color{blue}{(x^2+4x+6)}-\color{red}{(4a+16)(x)+(b-6a-60)}
\end{align*}
For $x^2+4x+6$ to divide the given polynomial.We need $4a+16=0$ and $b-6a-60=0$. Thus $a=-4$ and $b=36$.
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