Does this just proves that there are two positive real roots? How can I find these roots as a f[A,L]?

@J.D'Almbert I updated my answer. Root's are symbolic representations of exact solutions, under specific conditions one might apply ToRadicals to Root objects finding solutions in terms of radicals but in general you cannot expect such solutions because it is mathematically impossible to find solutions in terms of radicals for polynomials of order higher than $4$.

It is important to add that L>>A, e.g. when W=3 and A=2, L=8.7532759347618317962647585592473591641754313583030763

What is $W$?. Edit your question and make it clear since I'm not sure what you really want if my answer does not satisfy you needs.

When you simplify "sol" could you make A=2 and L=8.7532759347618317962647585592473591641754313583030763

Solve[p[x, 2, 8.753275] == 0 && x > 0, x, Reals] returns two solutions {{x -> 2.01469}, {x -> 3.00001}}.

Ok so that means the only solution I'm concerned with is the second one. Now, is it possible to represent this solution as just a f[A,L]?

I have done it in the answer, Root objects represetns solutions in terms of A and L, if you apply N you will find numerical approximation.

The only problem is I have been using a trial version of Mathematica so I would greatly appreciate it if you could apply N

If you expect any help you have to ask a precise question. I don't know what is W and nobody knows it. It doesn't matter at all what it is, but overall I don't know what you are looking for

I'm looking for a simple solution for W, which would be the second solution of x.

When A and L are symbolic you cannot expect more

than Root object

What other detail is needed?

there are solutions in terms of radicals only for 4-th order polynomials

Having such involved coefficients it is unlikely to expect that you will find a general representation in terms of radicals.

For example Solve[p[x, 0, 2] == 0 && x > 0, x] returns a double solution: {{x -> 2}, {x -> 2}}, then the polynomial is quite simple p[x, 0, 2] == 512 x^4 - 512 x^5 + 512 x^6 - 384 x^7 + 96 x^8

K so what your saying is that there is no representation for the second solution of x in terms of [L,A]

It is represented in terms of Root objects

however they are exact solutions

But u can see why that isn't really helpful because what that means is W is represented as a root object, and the variable of that root object can be represented as a root object

I said I don't know what is W

W is the solution of the octive, or the root.

and it does't matter

If you expect some clarification edit your question and make it precise

Can't I approximate the aforementioned solution of x

You can if you put numeric values for A and L, when they are symbolic then solutions are in terms of Root objects and there is nothing more to add

k

Read this answer How do I work with Root objects? for more

Its is said that " However even higher order algebraic equations can be solved explicitly if an associated Galois group is solvable ", can I use this idea?