Possible to enhance image for 300 DPI?

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I have a client who wants my photos sent at 300 DPI for printing. My photos are 800 x 600 resolution and from a bit of research I have come to the conclusion that these photos could be printed at an approximate maxiumum of 3x4 inchs @300dpi without loosing quality.

I was wondering if there were any resonable methods to increase my images resolution(without loosing quality), so they could be printed larger than my estimated 3x4 inchs while at 300dpi. For instance my friend mentioned something called smart resize on paintshop. I am aware that there are methods of increasing image resolutions alledgedly without distorting quality, but I am skeptical as to if these methods would work well for printing for a client, for example if they just mask pixelation on screen which might show up on printing.

Is it worth trying to edit these photos(making them a larger resolution without loosing quality). Or would it just make the images worse/ or having no significant effect on the size they can be printed decently @300dpi.

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at 800x600 size you only have that many pixels of information in the image at most (less if you are using a lossy format such as JPEG) and so at most your image can print at 2-2/3 x 2 inches at 300 dpi without losing quality as that is as much information as is there. To make the image any larger means that you are inventing information to fill the missing space as you increase the number of dots in the image beyond what was originally recorded. If the image is a JPEG or some other lossy format then chances are you already have less information in the image than that and any attempt to resize the image will lose further information.