Derivative Definition

Q. Let $f:R\to R$ be a function such that $f(x+y)=f\left(x\right)+f\left(y\right)+x^{2}y+x{y}^2$ $\forall x,y\in R$. If $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}=1$, then $f\left(x\right)$ is

1. $1+\frac{x^2}{2}$
2. $x-\frac{x^2}{3}$
3. $x+\frac{x^3}{3}$
4. $1+\frac{x^2}{3}$