<s>[INST] Regarding complex roots of a polynomial I am having difficulty in finding the roots of the following polynomial when it is given that all the roots are complex:-
$$f(x)= x^4+4x^3+8x^2+8x+4$$
How can I factorize the polynomial to get its roots?
 [/INST] Using binomial theorem;
$$\begin{align}f(x)&=x^4+4x^3+8x^2+8x+4\\
&=\left(x^4+4x^3+6x^2+4x+1\right)+2\left(x^2+2x+1\right)+1\\
&=(x+1)^4+2(x+1)^2+1\\
&=\left((x+1)^2+1\right)^2\\
&=\left(x^2+2x+2\right)^2\end{align}$$
</s>