Can a laptop with a mixer (like the mixpre) fare as well as an external recorder such as the tascam HD-p2 for recording great quality/noise-free sound?
With a quality audio interface, most definitely. There are some drawbacks due to the size and weight if you want to use it for run-and-gun style shooting but for traditional cart-based scenarios it works quite well and there have been a number of major motion picture releases recently that have been recorded that way. I would NOT just plug a MixPre into the inputs on the laptop's internal soundcard and expect top notch results though. A proper multichannel interface such as a MOTU 828 firewire interface is best. Sound Devices has a new USB interface with preamps that looks very promising, combining some enhanced features of the MixPre with better metering etc into one box with the A/D converters and USB I/O. For Mac laptops there are a couple of software packages available, Boom Recorder and Metacorder, that are purpose built for film and video recording, including such features as timecode, etc. For Windows you can use a number of programs - Adobe Audition, SoundForge, Vegas, etc
I love statements like "recording great quality/noise-free sound".
The hardware has something to do with it but not nearly as much as the people and choices they make do. Nothing is "noise free". The "noise" your going to pick up in a field situation that comes from your gear is almost always going to be swamped be the "noise" you didn't notice when you picked the location. "Great Quality" has a lot more to do with who's doing the recording than what you are recording with.
Once you get the location and production mixer/ boom op in order THEN you should start worrying about hardware.
To that point you want a mixer that can go into the laptop via firewire (or USB2) not an analog mixer that is going to have to feed the poor quality line in on a laptop. The good news is you can probably pick one up for a lot less than a MixPre. You need AC power (or some big batteries and an inverter) to keep the laptop rig going and it's really a cart setup, so no run and gun. But other than that it's a great option. If your on a Mac then BoomRecorder is what you want (and it's cheap - I think the two track version is still $20). Can't help you on the PC side.