British Prime Minister David Cameron planned to discuss the fight against extremism with President Xi Jinping on Thursday on a state visit hailed as a landmark by both China and Britain. Xi and his wife Peng Liyuan said goodbye to Queen Elizabeth II after spending two nights as the monarch’s guest at Buckingham Palace. Xi will dine at Cameron’s country estate, Chequers, before visiting the northwest England city of Manchester to end his four-day trip. Cameron’s office said Thursday’s talks would focus on international affairs and how China can help battle extremism. READ MORE: Everything you need to know about Xi's UK visit Western governments are keenly interested in China’s response to the war in Syria. China joined Russia to block a series of Western resolutions at the United Nations against Bashar Assad’s government. Russia has stepped up a military intervention in Syria that it says targets the Islamic State group but that Britain and others say seeks to prop up Assad. Peng earlier slipped away from the pomp, politics and deal-making of Xi’s state visit to meet teenagers studying the Mandarin language – and some of Britain’s pop culture icons. Xi’s wife, 52, a huge star in China with independent fame as a folk singer, spent the first few days of the visit, the first by a Chinese leader in 10 years, touring a succession of schools, events and universities in London. During a visit to a school in north London she listened to poetry and songs in Mandarin, widely considered one of the most difficult languages for English speakers. “Your Chinese is better than my English,” she told the students at Fortismere school, according to translated words. She also watched a Mandarin class during which students learnt Mao Zedong’s saying that “girls hold up half the sky”. Your Chinese is better than my English Peng Liyuan to students Asked by students whether girls and boys were equally represented in China’s education system, Peng said too many "”good girls” missed out on a place despite working harder than the boys. “It was an absolutely amazing experience, quite surreal at times,” the school’s headteacher Helen Anthony said of Peng’s visit. “She was delightful, warm, and she very much put the students at ease.” Xi has been greeted with elaborate British pomp and lavish ceremony on the state visit. Watch: China's Xi Jinping denies hard landing threat The two countries signed trade agreements worth more than 30 billion pounds (US$46 billion) – including a contentious nuclear power deal – as Cameron said Britain sought to be China’s “partner of choice” in the West. There have also been protests by human rights activists and pro-Tibet groups, and opposition politicians have urged Cameron to press Xi on China’s rights record.