The author studies the Hyers-Ulam stability of the functional equation
using the so called direct method of Hyers. A function is called a quartic mapping if it satisfies the above functional equation (FE). The author proves the following result: Let be a normed linear space and be a real complete normed linear space. If satisfies the inequality
for all with a constant (independent of and ), then there exists a unique quartic function such that . This result is obtained through six lemmas.