Yes, using a purely mechanical process every quintic can be solved in closed form this way. The usual method isn't all that different from the way the quartic i

Yes, using a purely mechanical process every quintic can be solved in closed form this way. The usual method isn't all that different from the way the quartic is solved. You do a change of variables to eliminate the x⁴ term, and then a few more simple changes of variables will get it to the Bring normal form, for which the solution is the Bring Radical. Then you can unwind the changes of variables to get a closed form expression in the original variable x. Each step is totally mechanical and involves no "trial and error" at all. The only issue is that the resulting expression can be extremely large once you unwind the change of variables. It's kind of similar to taking (for instance) a small C++ program and explicitly expanding every macro or function with a copy of the entire function body, producing a huge space increase. You are usually better off leaving things in terms of the transformed variable, along with the formulas to convert back to the each previous variable. This is also an issue with the solution to the general quartic, though, so it's neither here nor there regarding Bring radicals.