<s>[INST] If $x=2+i$, $gcd(a,b,c)=1$, and $ax^4+bx^3+cx^2+bx+a=0$, then what is $|c|$? 
Suppose
  $$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$
  where $a,b,c$ are integers whose greatest common divisor is $1$.
  Determine $|c|$.

So I first simplified the exponents and combined like terms.
I received $$a(-6+24i)+b(4+12i)+c(3+4i)=0.$$
I don't really know how to progress. What is the answer $|c|$?
 [/INST] We use your calculation. Note that as the other answers show there would have been nicer ways to proceed.
From the equation that you obtained we get $-6a+4b+3c=0$ and $24a+12b+4c=0$. Divide the second expression through by $4$, and add the first expression. We get $7b+4c=0$.
Thus $4c=-7b$ and therefore $c=7k$, $b=4k$ for some integer $k$. Substituting in the first equation we get $6a=5k$. It follows that $k=6m$ for some integer $m$. Thus $c=42m$, $b=-24m$, and $a=5m$. 
By relative primality we get $|m|=1$, and therefore $|c|=42$.
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