<s>[INST] Finding $k$ such that $(x^2 + kx + 1)$ is a factor of $(x^4 - 12 x^2 + 8 x + 3)$ 
$(x^2 + kx + 1)$ is a factor of $(x^4 - 12 x^2 + 8 x + 3)$ . Find $k$

....couldnt figure out how to find $k$
I tried assuming $(x^2 + kx + 1)= (x - 1)^2 $ where $k = (-2) $ comsidering that $(x-1) $ is a factor of the above polynomial.....but it didnt help either.
 [/INST] Let $x$ be a root of the polynomial $x^2+kx+1$, thus $x^2=-(kx+1)$. By substitute it into ${x}^{4}-12\,{x}^{2}+8\,x+3$ we get  ${x}^{4}-12\,{x}^{2}+8\,x+3=\left( -{k}^{3}+14\,k+8 \right) x-{k}^{2}+16 \equiv 0$.  Then solve system 
\begin{cases}
 -{k}^{3}+14\,k+8=0,\\
-{k}^{2}+16=0,
\end{cases}
we get that $k=4.$
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