<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] Write  $$x^4 - 12 x^2 + 8 x + 3=q(x)(x^2 + kx + 1)$$
then $q$ must be quadratic, so $q(x)= ax^2+bx+c$. By expanding and comparing the coeficients you will get $k$ (clearly $a=1$ and $c=3$)...  
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