<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] A polynomial of this symmetric form can be reduced in degree by the following trick. Divide by $x^2$ and get:
$\begin{align}
a (x^2 + x^{-2}) + b (x + x^{-1}) + c
  &= a (x + 1/x)^2 + b (x + 1/x) + c - 2 a
\end{align}$
The substitution $y = x + 1/x$ finishes this off.
This works because the powers of $x + 1/x$ turn out to be combinations of $x^k + x^{-k}$.
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