Can I patch two pieces of short pile carpet together?

Thank you in advance for your expert advice. My dog was sick on our landing carpet and a pro carpet cleaner was unable to remove the stains despite trying for over an hour. Luckily I have some spare carpet left over from when it was fitted last year that will cover the stained area but not the whole landing. The carpet is short pile as it runs up the stairs - if I were to cut the stained area out and replace it with the spare piece is there any possibility of the join being hidden or is that impossible for sort pile? Thank you!