I would try the card, since you already have it. Hook it up to your monitors and turn the sound way up but playing back nothing. If there is a noise there, it will probably be from other computer hardware electronics bleeding into the signal. That will be a limitation of the sound card, and the only way to fix it is to get a new card with good hardware.
IF you are using this for recording, or studio work, you will want to get a serious sound card. There is no way around that.
If you are just using this for computer speakers, and you are just listening to MP3s or playing games, than maybe that sound card won't make a big difference, depending on how noisy the signal is. Maybe, somehow, the shielding is good enough to let you have a good signal, if you are lucky. However, if there is noise there, it will mask the detail that studio monitors are really built for. Those speakers are made to play back everything and miss nothing. If you have a crappy line-out jack on your computer, you will certainly hear it, and what's worse is that you will hear it at the expense of the detail which would otherwise makes those speakers shine.