The black cover in that area is the timing cover.
The noise you are hearing is most likely coming from inside the timing cover. It could be coming from the power steering pump, a/c compressor or alternator but it sounds like the noise is probably from inside the timing cover. A quick way to rule out the engine driven accessories would be to remove the belts and start the engine, if the noise is still there then it's coming from inside the timing cover.
The noise could be a few things if it's coming from inside the cover. It's possible that the belt could be starting to shred and it is hitting the plastic.
More likely though you are probably hearing either the idler, tensioner or water pump
. The timing belt tensioner has a pulley with a bearing that can go bad, the timing belt idler has a bearing that can go bad, or the water pump bearings may have failed.
If the noise is definitely coming from inside the cover then they'll take the cover off and inspect everything for looseness and see if anything makes noise when spinning it by hand. It sounds like the noise is pretty bad so it shouldn't be hard to spot the failed component once the cover is off.
If the timing belt is more than a few years old most likely they'll recommend replacing it either way. The belt has a service life of 100,000 miles and it makes good sense to replace it while everything is apart even if undamaged. I would also consider replacing the water pump also, it is only another .5 hour in labor at that point, and if it failed in the future you would have to pay the labor again to go back in, plus it would damage the timing belt if it did leak.
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