On some equations stemming from quadrature rules

We deal with functional equations of the type $$F(y)-F(x) =(y-x)sum_{k=1}^nf_k((1-lambda_k)x+lambda_ky),$$ connected to quadrature rules and, in particular, we find the solutions of the following functional equation $$f(x)-f(y)=(x-y)[g(x)+h(x+2y)+h(2x+y)+g(y)].$$ We also present a solution of the Stamate type equation $$yf(x)-xf(y)=(x-y)[g(x)+h(x+2y)+h(2x+y)+g(y)].$$ All results are valid for functions acting on integral domains.