Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$ find all possible values of $k$.
so, $x^3 + \frac{b}{a}x^2 + \frac{c}{a}x + \frac{d}{a} =0$
$x^3 + \frac{c}{b}x^2 + \frac{d}{b}x + \frac{a}{b} = 0 $
subtracting the second equation from the first.
$(\frac{b}{a} - \frac{c}{b})x^2 + (\frac{c}{a} - \frac{d}{b})x + (\frac{d}{a} - \frac{a}{b}) = 0$
Discriminant
$(\frac{c}{a} - \frac{d}{b})^2 -4(\frac{b}{a} - \frac{c}{b})(\frac{d}{a} - \frac{a}{b}) = 0 $
$\frac{c^2 - 4bd}{a^2} + \frac{d^2-4ac}{b^2} + \frac{2cd}{ab} + 4 = 0$
don't have any ideas after this
HINT…You have both $$ak^3+bk^2+ck+d=0$$ and $$bk^3+ck^2+dk+a=0$$
Multiply the first equation by $k\neq0$ and subtract the second equation - what can you deduce about $k$?