The curve $C_1$ undergoes the transformations in the order given below:

1. A translation of 2 units in the negative $x$ direction.
2. A stretch parallel to the $x$ axis, factor 2, $y$ axis invariant.
3. A translation of 1 unit in the positive $y$ direction.

The resulting curve $C_2$ has equation
$y = \frac{x^2 + 9x + 22}{x + 4} , \quad x \in \mathbb{R}, \ x \ne -4.$ Find, in the simplest form, the equation for $C_1$.

1. Express the sequence of transformations acting on a general function for curve $C_1$.

Let the equation of $C_1$ be $y = \mathrm{f}(x)$.

1. A translation of 2 units in the negative $x$ direction replaces $x$ with $(x + 2)$:
   $y = \mathrm{f}(x + 2)$
2. A stretch parallel to the $x$ axis with factor 2 replaces $x$ with $\frac{x}{2}$:
   $y = \mathrm{f}\left(\frac{x}{2} + 2\right)$
3. A translation of 1 unit in the positive $y$ direction adds 1 to the function:
   $y = \mathrm{f}\left(\frac{x}{2} + 2\right) + 1$
   This resulting expression represents the equation of $C_2$.

2. Equate the transformed function to the given equation of $C_2$ and isolate the function part.

$\mathrm{f}\left(\frac{x}{2} + 2\right) + 1 = \frac{x^2 + 9x + 22}{x + 4}$
Subtract 1 from both sides to isolate the $f$ term:
$\mathrm{f}\left(\frac{x}{2} + 2\right) = \frac{x^2 + 9x + 22}{x + 4} - \frac{x + 4}{x + 4}$
$\mathrm{f}\left(\frac{x}{2} + 2\right) = \frac{x^2 + 8x + 18}{x + 4}$

3. Use an algebraic substitution to rewrite the equation in terms of a single variable, solving for $\mathrm{f}(u)$.

Let $u = \frac{x}{2} + 2$.
Solving for $x$, we get $\frac{x}{2} = u - 2 \implies x = 2u - 4$.
Substitute $x = 2u - 4$ into the isolated equation:
$\mathrm{f}(u) = \frac{(2u - 4)^2 + 8(2u - 4) + 18}{(2u - 4) + 4}$

4. Expand and simplify the algebraic fraction to find the simplest form of the equation for $C_1$.

$\mathrm{f}(u) = \frac{4u^2 - 16u + 16 + 16u - 32 + 18}{2u}$
$\mathrm{f}(u) = \frac{4u^2 + 2}{2u}$
$\mathrm{f}(u) = \frac{2u^2 + 1}{u}$
Replacing the dummy variable $u$ back with $x$, the equation for $C_1$ is:
$y = \frac{2x^2 + 1}{x}$