Dave you can, and I have, for many years been doing exactly that it is my trade.Of course it is not ideal and would never be a replacement for a backing that is totaly adhered originally.You would certainly not inject glue ( blimey be there all day )you would spray glue the backing to the carpet allow it to get tacky dry and then restretch it ,the carpet itself (not the backing ) will only stretch a certain amount as it would rip so it can more than happily be stuck back to the secondary backing restretched and recut without a problem regarding how much more of it there is . Regarding not worth the hastle mate well I and a lot of other fitters make a good living out of doing this as a part of our business.The only problem you would have in the stretch is in a doorway because it has stretched it could leave a gap ,but is certainly not a problem for a good fitter.Carpets are designed to stretch or they would wear out a lot quicker ,admitedly the cheaper types as shown are not as likely to look as good or wear as well and are harder to make good,but a great deal of carpets when first laid need a good restretch after a while and some do de laminate very early on in certain areas ,but are glued and refitted .I have glued delaminated carpets and restretched them and gone back years later and they are still ok .