Jose Mourinho’s side had started the season badly but the victory over Deportivo means they are eight points behind Barcelona.
As so often, Ronaldo was the hero, his hat trick and goals from Pepe and Angel di Maria earned Madrid the win.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.