Could you please see the attached battery images and tell me if it is safe to continue using this battery or should I discard it?
If the battery cost less than the thing you're powering, then you should probably discard the battery.
If you're stubborn (or very strapped for cash) and the battery didn't catch fire when the original damage happened, then it might perform okay, but I'd say there's a moderate chance of fire, especially at high currents.
Check battery voltage with a multimeter, then come back and check it again after a few hours. You're looking for a high self-discharge rate, which would indicate the battery is internally shorted out.
If the battery voltage drops noticeably after a few hours (or even a day), then you should trash the battery.
The terminal voltage on a lithium battery only drops about a quarter of a volt from 80% to 20% state of charge, so anything more than a hundredth of a volt or two and I'd call that significant. Every hundredth is about 2.4 percent of total battery capacity in that range, so 0.02V is roughly 5% of battery capacity.
Assessing damage from a picture isn't going to tell you anything concrete. If you're already considering discarding this battery, you might as well test its electrical properties (and physical properties) against an intact replacement.
At a minimum, you should check whether the damaged battery has more internal resistance (i.e. can't supply as much current), runs hotter, has more self-discharge, and/or has less capacity than an intact battery.
The safest thing to do is discard it.
I'm 14, please pardon me for any rookie sentences...
As far as my knowledge goes with Lithium Polymer batteries(not very far), they don't explode with a Hollywood style explosion, where flames go everywhere and everyone dies.
Instead, LiPo's start to smoke and the plastic cover on the outside layer of the LiPo will start to expand, when it cracks it releases hot steamy gas...
Judging by the damage, it looks like you dropped it... I don't this it's life threatening to use it... but I suggest you dispose it off, if you care about the electronics that are being powered by it...
Because the LiPo is 3 cell (11.1V) you can salvage working (undamaged) cells out. you will be able to use the undamaged cells for something else... you could also salvage the protective circuit (if the LiPo has it).