The curve $C$ has equation $y = \frac{x^2 + kx + 4}{x + 1}$, where $k$ is a constant.

1. Find the first derivative of $y$ with respect to $x$ using the quotient rule.

Let $y = \frac{x^2 + kx + 4}{x + 1}$.
Using the quotient rule:
$\frac{dy}{dx} = \frac{(2x + k)(x + 1) - (x^2 + kx + 4)(1)}{(x + 1)^2}$
$\frac{dy}{dx} = \frac{2x^2 + 2x + kx + k - x^2 - kx - 4}{(x + 1)^2}$
$\frac{dy}{dx} = \frac{x^2 + 2x + k - 4}{(x + 1)^2}$

2. Equate the derivative to zero and establish the condition for two distinct stationary points.

For stationary points, we set $\frac{dy}{dx} = 0$:
$\frac{x^2 + 2x + k - 4}{(x + 1)^2} = 0$
$x^2 + 2x + k - 4 = 0$

For the curve $C$ to have two distinct stationary points, this quadratic equation must have two distinct real roots (and the roots must not coincide with the vertical asymptote $x = -1$).

3. Apply the discriminant condition to find the corresponding range of $k$.

For two distinct real roots, the discriminant $\Delta > 0$:
$b^2 - 4ac > 0$
$(2)^2 - 4(1)(k - 4) > 0$
$4 - 4k + 16 > 0$
$20 - 4k > 0$
$4k < 20$
$k < 5$

(Note: Checking $x = -1$ in the numerator gives $(-1)^2 + 2(-1) + k - 4 = k - 5 = 0 \implies k = 5$. Since our range restricts $k < 5$, $x$ will never be $-1$. Thus, the roots are valid stationary points.)

1. Substitute $k = 4$ and rewrite the expression to find the equations of the asymptotes.

Substitute $k = 4$ into the equation of $C$:
$y = \frac{x^2 + 4x + 4}{x + 1}$
Perform algebraic division (or rewrite the numerator):
$y = \frac{(x+1)(x+3) + 1}{x + 1}$
$y = x + 3 + \frac{1}{x + 1}$
As $x \to \pm \infty$, $y \to x + 3$. Thus, the oblique asymptote is $y = x + 3$.
As $x \to -1$, $y \to \pm \infty$. Thus, the vertical asymptote is $x = -1$.

2. Find the coordinates of the axial-intercepts.

To find the $y$-intercept, substitute $x = 0$:
$y = \frac{0^2 + 4(0) + 4}{0 + 1} = 4$
The $y$-intercept is $(0, 4)$.

To find the $x$-intercept, substitute $y = 0$:
$x^2 + 4x + 4 = 0$
$(x + 2)^2 = 0 \implies x = -2$
The $x$-intercept is $(-2, 0)$.

3. Find the coordinates of the turning points.

Using the rewritten form $y = x + 3 + (x+1)^{-1}$, differentiate with respect to $x$:
$\frac{dy}{dx} = 1 - \frac{1}{(x + 1)^2}$
Set $\frac{dy}{dx} = 0$ to find stationary points:
$1 = \frac{1}{(x + 1)^2}$
$(x + 1)^2 = 1 \implies x + 1 = 1 \text{ or } x + 1 = -1$
$x = 0 \text{ or } x = -2$
Substitute back to find $y$-coordinates:
When $x = 0$, $y = 4$.
When $x = -2$, $y = 0$.
The turning points are $(0, 4)$ and $(-2, 0)$.

4. Sketch the graph of $C$ incorporating all the calculated features.

Draw axes and label the origin.
Draw the dotted line $x = -1$ and the dotted line $y = x + 3$, labelling them.
Plot the minimum turning point at $(0, 4)$ on the upper-right branch.
Plot the maximum turning point at $(-2, 0)$ on the lower-left branch.
Draw the two distinct branches of the hyperbola approaching the respective asymptotes.

1. Rearrange the given equation to relate it to the curve $C$ and a straight line.

Given the equation for $k = 4$:
$x^2 + 4x + 4 - (mx + 3)(x + 1) = 0$
Dividing by $(x + 1)$, we obtain:
$\frac{x^2 + 4x + 4}{x + 1} = mx + 3$
This equation models the $x$-coordinates of the intersections between the curve $C$: $y = \frac{x^2 + 4x + 4}{x + 1}$ and the straight line $L$: $y = mx + 3$.

2. Identify the geometric properties of the line $L$ relative to the sketch of $C$.

The line $y = mx + 3$ always passes through the point $(0, 3)$ with gradient $m$.
From the sketch in (ii), the oblique asymptote of $C$ is $y = x + 3$, which also passes through $(0, 3)$ and has a gradient of $1$.

3. Use the gradients from the sketch to deduce the range of $m$ for no intersections.

For the equation to have no real roots, the line $L$ must not intersect the curve $C$.
Pivoting around $(0, 3)$, if the line is steeper than the asymptote ($m > 1$), it will cross into the region containing the upper right branch of $C$ and intersect it.
If the line is less steep than or parallel to the asymptote ($m \le 1$), it remains in the empty region bounded between the two branches of $C$ and its asymptotes, thus avoiding intersection.
Since the question specifies $m$ is a positive real number ($m > 0$), the required range is:
$0 < m \le 1$