You're probably gonna need to replace all 4 - if one burst (unless it's 'cause you ran over something) then they're all probably old. At the VERY least replace the bad tire and the one on the opposite side of the car - otherwise you can have serious traction issues that can cause wrecks.
As to what tire you want, it all depends on the mission of your car. Are you looking for the ultimate in cornering ability in a street legal tire? (Falken Azenis) Are you looking for almost as good in the corners without sacrificing wet performance (Yokohama AVS ES100) or are you looking for a tire with good grip, but that will last kinda long? (Yokohama AVID).
The 512's you asked about have gotten good reviews, but you need to keep in mind that it's an all season tire. That means it delivers OK performance no matter what the weather, and that's why I don't have any personal experience with them - I run summer and winter tires. You won't get stunning grip out of 'em, and you won't get stunning snow handling out of 'em, but as an all around tire it seems pretty good.
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