Without the use of pexiderized versions of abstract polynomials theory, we show that on 2-divisible groups the functional equation
f\left( {x + y} \right) + g\left( {x + y} \right) + g\left( {x - y} \right) = f(x) + f(y) + 2g(x) + 2g(y)
forces the unknown functions f and g to be additive and quadratic, respectively, modulo a constant.
Motivated by the observation that the equation
f\left( {x + y} \right) + f({x^2}) = f(x) + f(y) + f({x^2})
implies both the additivity and multiplicativity of f, we deal also with the alienation phenomenon of equations in a single and several variables.