Since it's intermittent it will be difficult to troubleshoot and fix unfortunately. Takes 4 or 5 tries to start every time it has the failure you say? Next time it happens I would try removing the battery and unplugging it and then trying again. That should cause everything to completely reset. It would be an interesting experiment but I'm not sure that it will bring you any closer to a solution.
In terms of fixes you may want to make sure that your bios and video bios are both as up to date as possible but beyond that it sounds like an intermittent hardware failure.
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