The first step in solving a cubic equation is to apply a change of variables to reduce an equation of the form
x³ + bx² + cx + d = 0
to one of the form
y³ + py + q = 0.
This process can be carried further through Tschirnhausen transformations, a generalization of an idea going back to Ehrenfried Walther von Tschirnhaus in 1683.
For a polynomial of degree n > 4, a Tschirnhausen transformations is a rational change of variables
y = g(x) / h(x)
turning the equation
xn + an-1 xn-1 + an-2 xn-2 + … + a0 = 0
into
yn + bn-4 yn-4 + bn-5 yn-5 + … + b0 = 0
where the denominator h(x) of the transformation is not zero at any root of the original equation.
I believe the details of how to construct the transformations are in An essay on the resolution of equations by G. B. Jerrad.