1. The curve \(C\) has equation \(y = 2 \mathrm { e } ^ { x } + 3 x ^ { 2 } + 2\). The point \(A\) with coordinates \(( 0,4 )\) lies on \(C\). Find the equation of the tangent to \(C\) at \(A\).
2. Express \(\frac { x } { ( x + 1 ) ( x + 3 ) } + \frac { x + 12 } { x ^ { 2 } - 9 }\) as a single fraction in its simplest form.
3. The functions f and g are defined by

$$\begin{aligned} & \mathrm { f } : x \propto x ^ { 2 } - 2 x + 3 , x \in \mathbb { R } , 0 \leq x \leq 4
& \mathrm {~g} : x \propto \lambda x ^ { 2 } + 1 , \text { where } \lambda \text { is a constant, } x \in \mathbb { R } . \end{aligned}$$

1. Find the range of f.
2. Given that \(\operatorname { gf } ( 2 ) = 16\), find the value of \(\lambda\).