Determine the coordinates of points E and P, calculate the value of m and c, and calculate the length of NT.

Here's the solution to the problem:

4.1.1 Determine the coordinates of points E and P.
Points E and P are the x-intercepts of the graph $f(x) = -x^2 - 4x + 5$. To find them, set $f(x) = 0$:

$$-x^2 - 4x + 5 = 0$$

Multiply by -1 to make the leading coefficient positive:

$$x^2 + 4x - 5 = 0$$

Factor the quadratic equation:

$$(x+5)(x-1) = 0$$

This gives two possible x-values:

$$x = -5 \quad \text{or} \quad x = 1$$

From the graph, E is on the negative x-axis and P is on the positive x-axis.
Therefore, the coordinates are:

• E = $(-5, 0)$
• P = $(1, 0)$

4.1.2 Calculate the value of $m$ and $c$.
The line $g(x) = mx + c$ passes through points E and M.
From 4.1.1, E is $(-5, 0)$.
Point M is the y-intercept of $f(x)$. To find M, set $x=0$ in $f(x)$:

$$f(0) = -(0)^2 - 4(0) + 5 = 5$$

So, M is $(0, 5)$.

Now we have two points on the line $g(x)$: E$(-5, 0)$ and M$(0, 5)$.
Calculate the slope $m$:

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 0}{0 - (-5)} = \frac{5}{5} = 1$$

The value of $c$ is the y-intercept of the line, which is the y-coordinate of point M.

$$c = 5$$

Therefore, the values are:

• $m = 1$
• $c = 5$

4.1.3 Calculate the length of NT.
N is the turning point of $f(x) = -x^2 - 4x + 5$.
The x-coordinate of the turning point is given by $x = \frac{-b}{2a}$. For $f(x)$, $a=-1$ and $b=-4$.

$$x_N = \frac{-(-4)}{2(-1)} = \frac{4}{-2} = -2$$

Substitute $x_N = -2$ into $f(x)$ to find the y-coordinate:

$$y_N = f(-2) = -(-2)^2 - 4(-2) + 5 = -(4) + 8 + 5 = -4 + 8 + 5 = 9$$

So, the coordinates of N are $(-2, 9)$.

NTD is the axis of symmetry of $f$, which is the vertical line $x = -2$.
Point T is on the graph of $g(x)$ and on the axis of symmetry $x = -2$.
From 4.1.2, $g(x) = x + 5$.
Substitute $x = -2$ into $g(x)$ to find the y-coordinate of T:

$$y_T = g(-2) = -2 + 5 = 3$$

So, the coordinates of T are $(-2, 3)$.

To calculate the length of NT, since both points have the same x-coordinate, we find the absolute difference of their y-coordinates:

$$\text{Length NT} = |y_N - y_T| = |9 - 3| = 6$$

The length of NT is $\boxed{\text{6 units}}$.

4.1.4 Determine the coordinates of point S, the mirror image of point M with respect to the axis of symmetry.
Point M is $(0, 5)$.
The axis of symmetry is the line $x = -2$.
When a point is reflected across a vertical line, its y-coordinate remains the same. So, $y_S = y_M = 5$.
The x-coordinate of the axis of symmetry is the midpoint of the x-coordinates of M and S.
Let S be $(x_S, y_S)$.

$$\frac{x_M + x_S}{2} = -2$$ $$\frac{0 + x_S}{2} = -2$$ $$x_S = -4$$

Therefore, the coordinates of point S are $\boxed{(-4, 5)}$.