In The Odyssey, what was the main problem for the protagonist in chapters 7–10? How did he solve the problem?

The main problem for Odysseus is the question of how he will get home to Ithaca. In book 7, Odysseus has been away from home for 20 years: 10 years were spent fighting in Troy and 10 years were spent trying to get home from Troy. After many obstacles involving gods, monsters, and women, Odysseus has almost made it home. He is in the land of the Phaeacians and, after he reveals his identity to them, he proceeds to tell them the story of what happened to the great Odysseus in the 10 years following the fall of Troy.
In book 9, he tells the Phaeacians of his first significant obstacle in getting home: the land of the Lotus-Eaters. The island did not seem dangerous at first, and this is precisely the danger—the land is too welcoming. If you eat the food of this land, namely, the Lotus, “you lose your hope of home.” Odysseus is determined to get home, so he must drive his men from this seductive land. Next, they arrive at the land of the Cyclopes. Soon they are imprisoned by the Cyclops Polyphemus, who traps them in his cave. To escape, Odysseus and his men get the Cyclops drunk and then blind him. As they sail away, Odysseus mocks Polyphemus, and in the middle of his boasting, he reveals his identity. This revelation ends up creating great problems for Odysseus because Polyphemus is Poseidon’s son. In revenge for Odysseus’s actions, Poseidon spends the next 10 years tormenting Odysseus, trying as best he can to stop Odysseus from ever returning home.
In book 10, Odysseus continues his tale for the Phaeacians. He says that he almost made it home, but right before arriving in Ithaca, his own men opened up the bag of wind given to them by the god of winds, Aeolus. This blows them far from Ithaca and back to the island of Aeolus. Then, they have to deal with giant cannibals, the Laestrygonians, who destroy 11 of his 12 ships. He and his remaining companions end up on Circe’s island. Circe is a sorceress, who changes Odysseus’s men into animals. Fortunately, Odysseus’s greatest strength is his cleverness. He was the one who came up with the idea of the Trojan Horse, which allowed the Greeks to finally beat the Trojans after 10 years of war. This cleverness, along with help from Hermes, allows him to withstand Circe’s magic, and she ends up changing his crew back into men. Still, Odysseus falls victim to her seductive, Lotus-like charms, and only after a whole year passes are his men able to convince him to head back home. When he asks Circe for advice on how to get home, she tells him he must travel to the underworld and ask the prophet Tiresias for guidance on how to reach Ithaca.
Thus, his journey continues, the desire to head home driving Odysseus onward in the face of even greater dangers. This problem will not be “solved” until after he has finished telling the Phaeacians the entire story. Once he finishes his long tale, the Phaeacians agree to help Odysseus get back to Ithaca (book 13), and on Ithaca, he must deal with a very different set of dangers.