The part of the flower that can carry out photosynthesis is the calyx. The calyx is made up of sepals, which are leaf-like structures that protect the developing flower bud. While the other options listed (androecium, gynoecium, and corolla) are important parts of a flower, they do not have the ability to carry out photosynthesis.

The androecium refers to the male reproductive organs of a flower, including the stamens which produce pollen. The gynoecium refers to the female reproductive organs, including the pistil which contains the ovary, style, and stigma. The corolla refers to the petals, which are often brightly colored and attract pollinators.

Therefore, the correct answer is option 3, the calyx.