Context. We investigated a solar active-region filament eruption associated with a C6.6 class flare and a coronal mass ejection (CME) in NOAA active region 08858 on 2000 February 9. Aims. We aim to better understand the relationship between filament eruptions and the associated flares and CMEs. Methods. Using BBSO, SOHO/EIT, and TRACE observational data, we analyzed the process of the active-region filament eruption in the chromosphere and the corona. Using the SOHO/MDI magnetograms, we investigated the change of the magnetic fields in the photosphere. Using the GOES soft X-ray flux and the SOHO/LASCO images, we identified the flare and CME, which were associated with this active-region filament eruption.Results. The brightenings in the chromosphere are a precursor of the filament expansion. The eruption itself can be divided into four phases: In the initial phase, the intertwined bright and dark strands of the filament expand. Then, the bright strands are divided into three parts with different expansion velocity. Next, the erupting filament-carrying flux rope expands rapidly and combines with the lower part of the expanding bright strands. Finally, the filament erupts accompanied by other dark strands overlying the filament.The overlying magnetic loops and the expansion of the filament strands can change the direction of the eruption. Conclusions. The time delay between the velocity peaks of the filament and that of the two parts of the bright strands clearly demonstrates that the breakup of the bright loops tying on the filament into individual strands is important for its eruption. The eruption is a collection of multiple processes that are physically coupled rather than a single process.