<s>[INST] To find the square root of a polynomial My question is:
Find the value of $k$ such that $$4x^6 - 24x^5 + 20x^4 + 68x^3 -44x^2 - 40x + k$$ is a perfect square. 
hey all i have made an edit. Sorry for the inconvenience.
Any help to solve this question would be greatly appreciated.
 [/INST] If the goal is to find $k$ so that the polynomial is a perfect square, start by noting that it must be the square of a cubic polynomial:
$$4x^6 - 24x^5 + 20x^4 + 68x^3 -44x^2 - 40x + k=(ax^3+bx^2+cx+d)^2\;.$$
Clearly this immediately require that $a=2$. Now the square of $(2x^3+bx^2+cx+d)^2$ is
$$4x^6+4bx^5+(4c+b^2)x^4+(4d+2bc)x^3+(2bd+c^2)x^2+2cdx+d^2\;,$$
so you have the following system of equations:
$$\left\{\begin{align*}
&4b=-24\\
&4c+b^2=20\\
&4d+2bc=68\\
&2bd+c^2=-44\\
&2cd=-40\\
&d^2=k
\end{align*}\right.$$
Clearly $b=-6$; the second equation then allows you to find $c$, and you can then use the third, fourth, or fifth to find $d$ and then $k$. (To play safe, you should verify that the third, fourth, and fifth equations all yield the same value of $d$.)
</s>