start with a general cubic polynomial

G[x_]:=x^3+ A x^2 + B x + C

G[x - A/3]

Simplify[G[x-A/3]]

Collect[Simplify[G[x-A/3]],x]

and a solution!

(q+(q^2 - p^3)^(1/2))^(1/3)+(q-(q^2 - p^3)^(1/2))^(1/3)

p=d/3;q=e/2;(q+(q^2 - p^3)^(1/2))^(1/3)+(q-(q^2 - p^3)^(1/2))^(1/3)

d=6;e=9;%5

Clear[p,q,d,e]