چكيده فارسي :
In this paper, we first investigate solutions for the functional equations
$$
f(2x+y)+f(2x-y)=g(x+y)+g(x-y)+h(x)+e(y)\qquad(*)
$$
and
$$
f(2x+y)-f(2x-y)=g(x+y)-g(x-y)+o(y), \qquad(**)
$$
in which $ f, g, h, e, o:\mathbb{R} \to \mathbb{R}$ are functions with $e(0)=0$.
Then, using our results, we give the solutions of the functional equation (***)
$$
f_1(2x+y)+f_2(2x-y)=f_3(x+y)+f_4(x-y)+f_5(x)+f_6(y).
$$
As a special case, we deal with the quartic functional equation (****)
$$
f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+24f(x)-6f(y)
$$
where $f_i, f:\mathbb{R} \to \mathbb{R}$.
