You will find company names on nearly everything that you have in your product, all the way down to the capacitors on the circuit board. There is a Doctrine of First Sale that should allow you to use the player in your product. There are little licensing caveats that can create exceptions, but you should be okay.
Of course, building your own will be better, but it's probably a better idea to wait until you have some volume before investing time, effort and production dollars in that direction.
I'm not your attorney; my answer to your question includes assumptions. If you want me to be your attorney, I'm easy to find.
You can, and no doubt will have to, use lots of branded products in your device. The only components you'd use that would be generic would be nails and screws --and sometimes even those are branded. You don't need a re-seller's license for that, and you don't need to pay royalties, because copyright law's "1st sale" or "exhaustion" doctrine means that once you lawfully buy a copyrighted work, you're free to re-sell it or use it as you like.
But a video player isn't the creative type of work that's copyrighted anyway. Copyrights are reserved for works like songs, plays, novels, photographs, drawings, movies, etc. Not functional electronic devices, which are most likely patented and trademarked.
You don't need to worry about the patent on the product, but you do need to be careful not to market your product as having this off-the-shelf video player, so that consumers could mistakenly believe that the manufacturer of the video player is also the source of your product. That consumer confusion equates to trademark infringement.
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As noted by my colleagues, once a product branded with a trademark or covered by a patent is lawfully sold the owner of the trademark or patent no longer has the right to prevent the product owner from using [alone or within a new product] or reselling that product.
The key exception is as already stated: The product owner cannot make it appear to consumers that the trademark or patent owner somehow endorses or approves of the re-sale [in your case of the sale of the new product] or is associated in any way with the person or company who bought the product.
But this "exhaustion" of trademark and patent rights relies upon an important premise -- that is, the sale of the branded or patent-protected must have been lawful. A "cheap MP4 player" [perhaps from China?] may infringe some other company's patents covering technology included in the MP4 player. In fact, it would very likely infringe absent a license [ http://www.mpegla.com/main/default.aspx ]. If you install an infringing "cheap MP4 player" into your product then your product also infringes the patents covering the MP4 technology. In short, you must engage in due diligence to investigate your suppliers to ensure that their products are lawfully offered to the marketplace before you incorporate those products into your product.
You should discuss this project with your own intellectual property attorney.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
Daniel makes excellent points about patent exposure. Let me take it one step further. You need to perform a patent clearance analysis for your product---and that includes determining whether patents cover the use of an MP4player as a component in products such as your product. Patents cover not only products, but also methods for making and using them. Even if the MP4 player does not, itself, violate patents of others and you are free to use it under the "First Sale" doctrine, there may be patents that cover combining an MP4 device or a similar device into a product such as the one you are building.
Unfortunately, whenever you build a product like this, there are likely numerous relevant patents. Further, you need to take steps to assure that you are not suggesting to your customers that the manufacturers of components that you use endorse or are affiliated with your product In short, you need to work closely with IP counsel, and failure to do so could lead to financial disaster.
Great answers. You are asking the wrong questions. Reread the answers of Attorneys Ballard and Ross. You cannot brand your device to falsely imply some name brand is behind it. And, your service may infringe a patent of someone else. Without a patent clearance you are taking a risk. And, you really should be thinking about how you can get your own patent.An IP attorney is important for all those issues. Big companies use patent attorneys to clear new products. Small companies complain about patents as being bad if they belong to someone else. So do big companies if the patents belong to someone else.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.