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By real analysis I mean least upper bound principle and notion of limits of functions and sequences.

Specifically, is there any known proof which shows that any real polynomial can be written as a product of quadratic and linear irreducible real polynomials, using only real analysis (without assuming existence of complex numbers or using field extension, no double integration) ?

I am going to submit my proof on this to a journal and searching hard for possible instances in literature.

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    So...you *already* have a proof of that result **without** assuming complex numbers, field extensions and etc.? Well, send it to a journal: if they already know of a prior proof they will let you know. – 2017-02-27
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    I changed the title slightly: I replaced "real analytic" with "real variable" as I think that is your intention. – 2017-02-27
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    @DonAntonio Writing a math paper is a time consuming job and a ill- written paper can easily get rejected. Given that my vocation is in a completely different field, I want to be sure the effort is going to be meaningful. – 2017-02-27
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    @sobasu If you *really* have that proof you say then writing that paper is worth all the time and effort, **even** if afterwards it turns out somebody else already did something about it. For one, having a paper you can ask people to check ti find mistakes, inaccuracies, etc. – 2017-02-27
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    A real analytic function is one that is locally equal to a power series. That is the way the term is usually used in mathematics. "Real analytic" does not mean of, or pertaining to, real analysis. – 2017-02-27
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    @zhw Thanks, I see your point. – 2017-02-27
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    Here's what I'll do: I'll change "real variable" to "real analysis". – 2017-02-27
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    Even better, refers the field but avoids the possible confusion. Thanks – 2017-02-27

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I disagree with the advice of sending a paper to a journal before searching the relevant literature. It is almost guaranteed that a paper on the fundamental theorem of algebra (a very classical and well-studied topic) will be rejected if you do not include mention on previous proofs, and comparisons, explaining how your proof differs from them, etc. It is not the responsibility of the journal editors to do such a search in place of the author.

Anyway, already Gauss' proof from 1815 is purely within real analysis, and basic algebra (not involving field extensions). You can see a translation here. There are several other more recent proofs that avoid complex analysis as well.

Again, the value of a paper on this subject would be to highlight the differences with previous approaches, or maybe a modern, streamlined presentation of a classical and not easily accessible or not very well-known proof.

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    Gauss's proof used "indeterminates", which are essentially complex numbers. There are fully real analysis proofs, by [Pukhlikov and Pushkar](https://www.google.ch/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwiqtJ6rrLjSAhVDuhQKHerhAwIQFggcMAA&url=http%3A%2F%2Fwww.math.uconn.edu%2F~kconrad%2Fblurbs%2Ffundthmalg%2Fpropermaps.pdf&usg=AFQjCNFnzjMNjmOTjaKsw1kUqjt3bN_ymg&sig2=h_ndhuEjMKiN3dEy8CSf1A&cad=rja), but they use quite involved topological machinery. Many proofs avoid complex analysis, but I could not find anything else which avoids assuming the existence of complex numbers. – 2017-03-02