Let $a$ be a real parameter such that $$f_a(x)= x^4-6x^3+11ax^2-3(2a^2+3a-3)x+1$$ has has four distinct complex roots, that form a parallelogram when plotted on the Argand diagram.
Prove That $f(x+z)$ has $4$ roots $\pm \alpha$ and $\pm \beta$, where $z$ is the center of the parallelogram.
I need some help with the question. I can't proceed.
Thanks in advance.
Notice that $x=c$ is a root of $f(x)$ iff $x=c-z$ is a root of $f(x+z)$. So the roots of $f(x+z)$ are just the points $c-z$ where $c$ can be any of the vertices of our parallelogram. But if $c_1$ is a vertex of the parallelogram and $c_2$ is the opposite vertex, then $z=(c_1+c_2)/2$. Thus $c_2-z=-(c_1-z)$. That is, if $\alpha=c_1-z$ is a root of $f(x+z)$, then $-\alpha=c_2-z$ is another root. This means that the roots of $f(x+z)$ come in two pairs $\pm\alpha$ and $\pm\beta$.