functional equations - Find all real functions that satisfy $f(xy+x)+f(y)=f(xy+y)+f(x)$

Can you please help me solve this problem: Find all real functions that satisfy the functional equation $f(xy+x)+f(y)=f(xy+y)+f(x)$? I should solve it using the following theorem:

Let $f\colon R\rightarrow R$ satisfy the Hosszu functional equation
$$f(x+y-xy)+f(xy)=f(x)+f(y)$$

for all $x,y \in R$. Then there exists an additive function $A\colon R \rightarrow R$ and a constant $a\in R$ such that $$f(x)=A(x)+a.$$

Any advice would be helpful. I tried putting $x=y=1$ but didn't know what to do next.