With which of the following do amoeba engulfs the food? 

Amoeba engulfs their food using pseudopodia, which are temporary extensions of the cell membrane that can be used for movement and phagocytosis. 

Pseudopodia are finger-like projections that help amoeba in catching food and also in locomotion. 

When amoeba encounters food particles, it extends their pseudopodia around the food particle and encloses it in a small pocket of the cell membrane, forming a structure called a food vacuole. 

The trapped food forms a food vacuole within which the digestion of food takes place. Therefore, the correct answer is "Pseudopodia."