Who was the first to show that this quintic equation has five radical roots?

Question

Despite this is being a very trivial question, I would like to know who was the first (with reference evidence) to show that the following quintic equation has five radical roots: $$x^5 + x + 1 = 0$$

What brings up this question is that "the radical real root for this specific example is impossible to be contained in a much wider range formula given as an independent following question for any polynomial degree as a trinomial polynomial (with all coefficients are zeros or (+/-) one (for only three terms),

however, that question which is an answer itself may not have been received well enough by some mathematicians mentioning that I'm not a professional mathematician which makes hard enough to convey the simplest idea to the professionals in this field

"Is this a NEW CORRECT One real approximated root in a rational form of the following $n$-th degree polynomial equation $$ x^n + x^m = 1,$$ where $(n > m)$ are two distinct positive integers, is given by the following series formula as an approximation where no real root ever exists exactly as we may imagine with our imaginative roots that are strictly associated or derived from that real root that never exists

$$ \begin{align} x &= 1-\frac{1}{n}+\sum_{k=2}^N \frac{(-1)^k \prod_{i=1}^{k-1}(km-in+1)}{k!n^k} \\ &= 1-\frac{1}{n}+\frac{2m-n+1}{2!n^2}- \frac{(3m-2n+1)(3m-n+1)}{3!n^3} +\frac{(4m-3n+1)(4m-2n+1)(4m-n+1)}{4!n^4} -\frac{(5m-n+1)(5m-2n+1)(5m-3n+1)(5m-4n+1)}{5!n^5} +\dotsb \end{align}$$

Where $N$ is a relatively small suitable chosen integer up to our practical needs and capability of making our real roots more sensible (only in rational form) for any other purpose since it is absolutely and perpetually impossible to make the sum of our terms go with no number like infinity since then our real root in our minds would simply become like a ratio of two no existing numbers (each with an uncountable number of digits) which isn't any number, especially that root also is impossible to be constructed exactly by any means

Note that in very rare cases, the above trinomial polynomial has real existing roots in constructible number form and strictly in their surd form (and never in their approximations in decimal or rational forms), when $(n = 2m)$, and $m$ is a power of two

REFERENCE:

https://www.quora.com/What-are-the-ways-to-understand-the-proof-that-there-is-no-formula-for-expressing-the-roots-of-the-general-quintic-equation-via-radicals/answer/Bassam-Karzeddin-1

For a reference to the more general solution of this Trinomial equation $$ax^n +bx^m +c = 0$$ click, OK, then click National then type in the search button "solution of equations by power series" at this link http://opac.nl.gov.jo/uhtbin/cgisirsi.exe/zJ7a8B3i0n/MAIN/263930002/123

Answer 1

I originally answered this on a different forum*. BK's solution (with some minor improvements by GW) is:

$$x=\sum_{i\geq 0}\frac{(-1)^{i}}{i! (i m+1)}\prod_{j=0}^{i-1}(\frac{i m+1}{n}-j).$$

It appears to be the first discovered general formula for real roots of the polynomial $x^n + x^m - 1 = 0$. It is not clear whether other radical roots can also be obtained using hypergeometric series.

*I would be interested in a discussion of mathematics forums and how they could be improved, and also how they could coordinate to address different kinds of questions.

I did find a "meta" question here about questions like: "Who first discovered X?". I couldn't participate in that discussion.

The reason for deleting BK's original answer (according to comment) is because it was not on topic of history of mathematics.