Dense set in positive real

[sdsert]

i, Let $f=x^2+2x$ be a quadratic polynomial and $\mathcal{R}_f$ be its range.

Denote $\mathcal{A}_{i+1}$ be the set of real numbers that lie in $\mathcal{R}_f$ get from finding roots of equation $f(x)=n_i$ where $n_i\in\mathcal{A}_i$, $\mathcal{A}_0=\mathbb{Z}\cap \mathcal{R}_f$.

Let $A=\mathbb{Z}\cup (\bigcup_{i=0}^{\infty} \mathcal{A}_i)$ and $B=\{x|x\ge 0,x\in A\}$. Is $B$ dense in $\mathbb{R^+}$ ?

ii, Find all quadratic polynomials $f$ such that $B$ is dense in $\mathbb{R^+}$.