With four games remaining, 17th-placed
Boro are in 19th spot on 24 points and
But the Tigers took the lead just past the hour when
They have now won four consecutive home league games.
Spanish striker Fernando Llorente gave
Time is running out for Boro who have not won in 16 league games, and they endured a nightmare opening 20 minutes in which they conceded two goals and had midfielder Gaston Ramirez sent off for a second booking.
Josh King scored just 96 seconds into the match, and Benik Afobe doubled the advantage for the Cherries.
Winger Marc Push made it 3-0 and Charlie Daniels completed the rout.
Meanwhile, West Ham and Everton played out a 0-0 draw in a desperately poor Premier League encounter at the London Stadium on Saturday.
It took 28 minutes for the first effort on goal, and recalled Everton goalkeeper Maarten Stekelenburg saved comfortably from Havard Nordtveit’s shot from outside the box.
Visiting manager Ronald Koeman made a double change at the break, introducing Gareth Barry and Ademola Lookman, and the tempo improved in the second period.Lookman forced two opportunities in quick succession and put both wide, while Stekelenburg saved from Manuel Lanzini, but there were few nervous moments for either side.