A Shared Tangent, Then a Bound

Let $f(x) = x^2 + ax + b$ and $g(x) = \mathrm{e}^x(cx + d)$. The curves $y = f(x)$ and $y = g(x)$ both pass through the point $P(0, 2)$ and have the same tangent line $y = 4x + 2$ at $P$.

1. Find $a, b, c, d$. 2. If $f(x) \leqslant kg(x)$ for all $x \geqslant -2$, find the range of possible values of $k$.