300 dpi download?

Hi, 

If I get download a firefly image, it gives me 2304 x 1792 pixel image, with a res at 72. the size of the image, is some 28 x  32.. which is great.

BUT, . Printers want a 300dpi image, so I incease the resolution. The image size shrinks to a mere 7.8 by 5 whatever inches. Which is a tiny size for an art piece. 

I also want to incorporate firefly images into my other art pieces, but cannot do as the size is small. 

How do I fix this when I create something?

Help, please and thank you.