<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] Observe that there is no cubic term, and the independent term is $3$. Hence the factorization must be
$$(x^2+kx+1)(x^2-kx+3)=x^4+(4-k^2)x^2+2kx+3.$$
By identification $k=4$.

The "standard" solution is by equating the remainder of the long division,
$$(8+14k-k^3)x+16-k^2$$ to zero. The only solution is $k=4$.
</s>