A Root and a Rational Bound

Let $f(x) = \ln(x+1) + \sqrt{x+1} + ax + b$, where $a, b \in \mathbb{R}$ are constants, and suppose the curve $y = f(x)$ is tangent to the line $y = \dfrac{3}{2}x$ at the point $(0, 0)$.

1. Find $a$ and $b$. 2. Prove that for $0 < x < 2$,