The transformation relating two quintic polynomials

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I apologise if this a silly question. Is there a transformation relating the following two graphs:

  1. $x^5 + -3 \sqrt{\frac{e}{5}}x^3 + ex + f$
  2. $x^5 + ex + f$

if so is there any way to derive it like some sort of specialised tschirnhaus transformation. Help would be appreciated.