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The tourists have already lost the series Down Under, but they gave themselves a strong start in their innings after posting a record score of 316-8.
Despite some positive bowling from England, Finch continued to frustrate the field and up the run-rate for the hosts, but England took control once again when Ben Stokes ended George Bailey's time with the bat.
The umpire failed to call the wicket, which prompted England to take a review for a potential caught-behind on the left side, and their decision proved successful as the review showed an edge, which sent Bailey packing.
Despite losing three wickets, Finch went from strength to strength by reaching his century from 97 deliveries, but Steven Smith (19) didn't last long at the crease as Stuart Broad caught the edge with his delivery.
The biggest wicket of the day came when Bresnan dismissed Finch for 108 when he guided the ball straight into Broad's hands on the third-man boundary.
Wickets then began to tumble for Australia as England wrapped up the match by bowling the Aussies all out for 259.