Final answer:
Béla Bartók is the classical composer known for incorporating folk melodies into his compositions, developing a musical style that reflected both modern and nationalistic elements by weaving in elements of Eastern European folk traditions.
Step-by-step explanation:
The classical composer known for incorporating folk melodies into his compositions is Béla Bartók. Unlike Ludwig van Beethoven, Wolfgang Amadeus Mozart, and Johann Sebastian Bach, who were associated with the Classical and early Romantic eras, Béla Bartók was a Hungarian composer and ethnomusicologist who is often noted for his collection and analytical study of folk music. Bartók turned to folk music as a means of developing a musical language that was simultaneously modern, nationalistic, and universal. His works are peppered with the rhythms, harmonies, and melodies of the Hungarian and other Eastern European folk traditions that he studied.