Find all functions
![f\colon\mathbb{R} \rightarrow\mathbb{R}](/media/m/5/4/7/5473961bdae24a3adf8c52f7cb678649.png)
satisfying the equation
![f\left(x^2 + y^2 + 2f\left(xy\right)\right) = \left(f\left(x + y\right)\right)^2](/media/m/6/4/c/64c0e8e5bc36e5fc9668ad2b1fe092ed.png)
for all
![x,y\in \mathbb{R}](/media/m/a/9/9/a99b34501d763183cdc80077dc114d15.png)
.
%V0
Find all functions $f\colon\mathbb{R} \rightarrow\mathbb{R}$ satisfying the equation $$f\left(x^2 + y^2 + 2f\left(xy\right)\right) = \left(f\left(x + y\right)\right)^2$$ for all $x,y\in \mathbb{R}$.