Salah is the first Liverpool player to get to 20 goals in a season by December since Ian Rush in 1986, and showed once more against Bournemouth why he is proving to be the best signing of last summer.
Dejan Lovren doubled the lead with a close-range diving header six minutes later and Salah made it 3-0 a minute before halftime by skipping past fullback Charlie Daniels in the right channel and beating goalkeeper Asmir Begovic with a neat left-foot shot.
Firmino hooked the ball back at the far post following a corner, for Lovren to dive in and head home for the second.
"We scored a fantastic first goal... we could have scored more".
Phil is a Liverpool player now and he is doing really well for us, he's been the most professional player for us all season.More news: The Ashes: Cook's struggles continue as Australia hone in on the urn
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- Philippe Coutinho has been directly involved in 15 goals in his last 10 away games for Liverpool in all competitions (8 goals, 7 assists).
He gives us his all every game and that's all you can ask of him.
Two down with only 26 minutes on the clock, Bournemouth were already on the ropes.
Bournemouth were offered a route back into the match with the score at 2-0, but Jermain Defoe's low strike hit the post and ricocheted to safety.
Alex Oxlade-Chamberlain nearly added to the score with a shot which cannoned back off a post in the second half before Firmino glanced home a header from another excellent Coutinho cross to confirm victory. I try to enjoy the moment.
Bournemouth manager Eddie Howe said his players "were well off it".
"But we are in good form and we will be going there full of confidence and looking for three points. We are well aware our style can make us vulnerable".