How can I fix the bump in my carpet? Please see two images below
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Carpets usually expand over time.
This is caused by the fabric, under felt and entire carpet construction to stretch, caused by people walking over it over extended periods of time.
It does not usually stretch as much as you show in your pictures. Possibly this happened because the material is sensitive to moisture and caused it to become more elastic. Moisture could be from spillage, leakage or frequent carpet washing.
Tools and techniques
If your carpet has been like that for a long time there could be irreversible damage. You could still see a crease or uneven surface after re stretching it.
Here is a nice tool (maybe you can rent one?) made specially for stretch repair.
But the more common tool is the kick stretcher and requires some elbow great!
How to do it?
Too much pressure from either tool could tear the carpet. So be careful. If in doubt, call a professional.
In case your carpet is loose
I just had another look at it looks like your carpet is more like a large rug? :)
In that case, where you do not have grippers on one side you could opt into buying some hard underlay.
Remove the entire carpet aka rug and glue the underlay to the carpet. I would not do it inside the home because it requires a special type of glue that really stinks and needs about 24 hours to bond properly, in a fairly warm and dry environment.
Thanks to the underlay, depending on what you choose your carpet will be warmer to the foot but also help keep it flat.
You need to restretch the carpet. It can be a do-it-yourself job for someone that likes to buy tools, but unless you already have them you won't save a lot. Though you might be able to rent one from a place that sells carpet retail, like (in the US) Home Depot.
Here is a page that shows how to do it, but if that link disappears, just search "carpet restretching."