galois theory – Polynomial equation with perfect squares powers

I am currently trying to solve these kinds of equations, such as:


How can I make an approach to these kinds of questions?
What have I tried:

Solving $x^9+3x^4-5x+7=0$
Assume that the equation is
Or something like that, then the $x^9$ will cancel but the problem is to solve the remaining $a, b, c$

If there’s is a quicker way feel free to answer me. I have heard of the Galois theory and if that helps, please write a detaided solution.