Quartic polynomials of a complex variable I want to answer the following:

The equation 
  $$
    a \bigg(z + \frac{1}{z}\bigg)^2
    + b \bigg(z + \frac{1}{z}\bigg)
    + c = 0
$$
  has four solutions. Find which quartics can be put in this form after apply a linear change of vars.


$a(z+\frac{1}{z})^2+b(z+\frac{1}{z})+c = 0$
This is a quartic polynomial, if we multipy by $z^2$ we obtain
$az^4+bz^3+(2a+c)z^2+bz+a = 0$
This quartic can be related to the general quartic polynomial
$f(x) = Ax^4+Bx^3+Cx^2+Dx+E$
Now compare the coefficient of the two polynomials
$az^4+bz^3+(2a+c)z^2+bz+a$ and $Ax^4+Bx^3+Cx^2+Dx+E$
Are you looking for a way to transform the general quartic to this form $az^4+bz^3+(2a+c)z^2+bz+a$ ?
Now we are looking for a quartic that can be put in that form, since it has $3$ variables $a,b,c$, then reduce the polynomials and compare with a depressed quartic $y^4+py^2+qy+r$, equate their coefficient then put $a,b,c$ in terms of the order coefficient
