2008 Sirion possible battery issue

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Hi all!

Looking to get some help/advise please.

My girlfriend has a 2008 Sirion which she recently inherited. Up until now there hasn't really been any issues with the car but last month the exhaust was replaced and now the car won't start. Went to use the car on Saturday and it was completely dead (didn't turn over, or make any noise. Dash did not light up.) Nothing in the car was turning on and the central lock stopped working so I guess it's the battery.

Attempted to jump start it using my car and after a few minutes I noticed the red light on the dash began to blink. Waited for about 10 mins with the jump leads attached and then attempted to start the car, this time the dash lit up but the engine still doesn't start and after a few attempts the power seems to go again and we're back to square one.

The battery it's self doesn't look that old which is what worries me, there is also no paper work to say when this battery was fitted however it has '11/07/17' written on the battery and interestingly there is paperwork for everything else including the last battery which was replaced in 2013. I did notice the terminals had huge lumps of grease on them which I'm thinking was put on there to stop corrosion? I removed a bit of the grease and tried again with the jump leads in for an hour, again got a little more signs of life (the stereo came on which I promptly turned off) but the engine still didn't start.

Am I right by thinking if it was the alternator I'd still be able to jump start the car? If anyone has any advise they could offer, I'd be very grateful.

Thanks in advance!