please focus on the concept to solve this problem, because i can't handle to research on diffcult terminology.Thanks in advance.
Find all real roots by galois theory and find all other root to this equation: $x^8+x^6+x^4=340$
If we can solve $$u^4+u^3+u^2=340$$ then by letting $u=x^2$ we can solve the original equation. But the displayed equation is of degree 4, and there is a formula for solving those. Just search for "quartic formula".
I'm sorry that this doesn't use Galois Theory, but I don't see what Galois Thoery can do for this problem.