Since the speaker sounds passive(does not have its own amp). Yes you can connect it right upto the card amplifier. You do NOT have to open it, inside it just has a short wire from those press clips to the speaker. so + red and - black and you are good to go.
Now for the part that may not be so good.
Many car speakers are 4 ohms and some are even lower(I have seen 8 ohms in a car, but not very often. You can check with a multi-meter). This is done because cars have a lower voltage to work with. Many home speakers are 8 ohm(some may be 6 and older ones may be 16 ohms).
Why does this matter? Because at the same output voltage a 4 ohm speaker will pull 2x the power of an 8 ohm speaker. So if the stock speakers are 4 ohms and getting 40 watts, the 8 ohm speaker will only get 20 watts at the same voltage.
This is because of something called ohms law. This would be a wall of text, but v = current(amps. normally A, but for ohms law it is I) over resistance and wattage is voltage x current(amps)
Here is a cheat.
http/security.livewatch.com/forum-ref/ohms-law-calculator
Speakers are not quite like resistors and audio is rarely running at peak voltage it is all over since it moves the speaker forward and back[it is AC not DC so to speak], but the fact that lower resistance = more wattage still holds true. The 8 ohm speaker would be very easy on your amplifier too since it is made for 4 ohm speakers.
I am not 100% sure how much a home sub like bumps on the road either, but it is a speaker(so it is kind of use to movement).