A Parabola Meets an Exponential

Let $f(x) = m(x - 2m)(x + m + 1)$ and $g(x) = e^x - 1$. Determine, with proof, which of the following are true:

1. when $m = 1$, the equation $f(x) = g(x)$ has exactly one real solution; 2. when $m \in (-1, 0)$: for every $x \in \mathbb{R}$, $f(x) < 0$ or $g(x) < 0$; 3. when $m \in (0, 1)$: for every $x \in (-\infty, -2)$, $f(x)g(x) < 0$; 4. there exists $m \in \mathbb{R}$ such that $f(x) - g(x) < 0$ for every $x \in \mathbb{R}$.