Why was Rome able to conquer Carthage and then go on to extend its influence across the whole Mediterranean basin and much of Western Europe?

Answer 1

By defeating Carthage, Rome gained control over the western basin of the Mediterranean. She did not gain control over the eastern Mediterranean.

In the First Punic War (264-241 BC) Rome defeated Carthage in Sicily and gained control over Sicily, Corsica and Sardinia.

In the Second Punic War (218-201 BC) Rome repelled an invasion of Italy by Hannibal, a Carthaginian general, and defeated the Carthaginians in southern Spain and took over their possessions there. Carthage was left with only her homeland territory (Tunisia and western Libya).

Rome fought the Third Punic War (149-146 BC) because she wanted to destroy Carthage, which she did.
Through three wars.

Answer 2

Rome established control over Italy, which gave it the military might to defeat Carthage three times in the three Punic Wars between these two cities. She had developed a system of alliances with some of the Italian peoples, who provided troops which supported the Roman legions. They provided at least 60% of the military manpower available to Rome. As a result, Rome had what was probably the biggest pool of military manpower in the Mediterranean. This was one of the main reasons why Hannibal failed to conquer Italy and seize Rome. The first of the three wars also led to Rome (which did not have a fleet to speak of) developing a navy. During the second war, the Roman fleet proved to be more powerful than the famed Carthaginian fleet.

Rome's victories in the first and second wars weakened Carthage. At the end of and soon after the first war Rome took over western Sicily, Sardinian and Corsica, which belonged to the enemy. Carthage lost the hub of her trading networks in the western Mediterranean. She responded by conquering southern Spain and the greater part of its east coast. However, Rome defeated Carthage in Spain in the second war. Carthage lost her territories there. Hannibal lost his power base in Spain and the military manpower he drew from there for his campaign in Italy. Carthage lost the silver mines in the Sierra Morena. Carthage became confined to her homeland territory in Tunisia and western Libya. She was no longer able to measure up to the growing Roman power. When the war party in Rome won the argument in favour of the destruction of Carthage, claiming that she was building up her military might again, the greatly weakened Carthage was doomed. Even so, Rome did not fare well in the early part of this war (the third one). She achieved her objective when she found a capable commander in Scipio Aemilianus .