An active splitter like the NAS LMX-82 - from National Antenna Systems?
Splits the input 2 ways, each way feeding a seperate pre-amp contained in the NAS, which then splits into 4 outputs x 2.
Depending on the coax type you use from the NAS LMX-82 outputs to each room, and the length those runs are going to be, as well as the coax length between the antenna and NAS, you may well still land up with very weak signals from the NAS, despite the fact it has 2 pre-amps in it.
Your solution then is going to be to boost the signal level coming down the coax to the NAS LMX-82, with a used 50 Ohm preamp (sourced on eBay?) located between the NAS and your antenna.
6dB - 9dB from that pre-amp would be plenty sufficient Gain - go much more than that and I see the pre-amps in the NAS module been pushed to saturation (which you do not want happening in a 50Ohm-into-75Ohm setup).