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If the damage from the nails caused structural problems with the tire the belts can separate, and the tire can be delamination, which can end in a blowout. The best way to check for this is to jack the vehicle up and spin the tire by hand. Hold your hand on the tread and feel for lumps in the tread, or other unevenness. If the tire seems perfectly round with no deformations then it will be fine and shouldn't have any structural issues. I would also suggest getting the balance of the tires checked, a wheel weight could have come off. You especially need to check the tire that was repaired. Patches although light could alter the balance, as well as the position the tire was remounted in.
2003 Jetta left rear window blue plastic part of the