The most likely situation is that the ESC needs to be callibrated, but that may not be the full extent of the problem. Hopefully you should have some instructions for accomplishing the calibration that came with the car or the esc. It's possible something is faulty, but let's do a proper diagnosis first.

Additoinally, from your (very good) description of the car's behaviour, it sounds like the brushless motor is connected incorrectly (even if you matched cable colours, this may be a factory error). There are two ways to reverse the direction of a channel, either on the transmitter with the 'reverse channel function', or by simply swapping ANY two neighboring motor wires (b with a or c for example). I believe you need to do both of these corrective actions. So, in steps in this order...

1) Reverse the throttle/brake channel on the transmitter.
2) Swap any two neighboring motor wires on the ESC (b with a or c for example).
3) Calibrate the ESC accroding to instructions.