even if I connect the arduino to the circuit board found in an electric-driven RC car?
That chip in the lower-left corner of the receiver board is the RX2 half of the TX2/RX2 chipset pair; a very commonly used and documented chipset for low cost (non-hobby grade) R/C toy cars.
You can do your project fairly easily with this chip; really you only have to take a few voltage measurements, and refer to the documentation of the chipset, to interface with it (well, actually, you don't interface -with- it, you actually bypass it - in fact, the chip can be completely removed, and actually should be, from the circuit). See the following lengthy thread on how to do this:
As DuaneB has noted, though, you can go the route of using a fully proportionally controlled R/C car (hobby-grade); this route has advantages to the "hack a cheap R/C car method"; namely, interfacing is much simpler (you just need to interface using the Servo library). The disadvantage to the method, though, is the fact that such a car tends to be much more expensive (and complex, mechanically - depending on if you have to completely assemble it from a kit) than a cheap R/C car. The cost difference is even greater, of course, if you already have the cheap R/C car (which you apparently do) or if you purchase one from a resale/thrift store (where many of these kinds of cars can be found for very little money).
So, this route you would gain ease of interfacing (Servo library) and proportional control of steering, in exchange for a higher up-front cost. While with the former method, you have to do more work on the interfacing end (write your own control routines, and do some soldering, etc), and no proportional steering (because most of those cars only have left/right/center control of steering), in exchange for a lower up-front cost.
Since your project doesn't -seem- to need Arduino control of proportional steering, and you can effect proportional control (ie, speed) of the h-bridge you would interface to on the cheap R/C car via PWM control, then the only benefit with going with the higher-cost option is ease of interfacing. Also note that if you absolutely had to have proportional steering of the cheap R/C car, you might find it cheaper to do a "hybrid" approach, and simply replace the simple actuator of the cheap car with an R/C servo, while continuing to use the R/C receiver board's on-board h-bridge to control the speed of the drive motor; versus buying an entire hobby-grade R/C car.
Finally - DuaneB: Help me out here in these discussions; you were a part of the TX2/RX2 chipset discussion that I linked (and provided a ton of assistance in) above - in the future on discussions about this kind of interfacing, especially if it is easily apparent that the poster has such a vehicle, provide a link to the discussion. I understand that you have a bias toward hobby-grade R/C cars for Arduino interfacing, but many people don't have the experience or budget to afford such vehicles, depending on the project at hand. Let us both continue to give these people unbiased information about both systems, so they can decide which would be best for their application.