The question asks you to justify why the two lines $g$ and $h$ are parallel. Here is how we can analyze it step by step:

1. Equation of Line $g$: $g : y = -\frac{x}{4} + 8$ The slope $m$ of the line $g$ is $m = -\frac{1}{4}$.

2. Parametric Equation of Line $h$: $h : X = \begin{pmatrix} 4 \\ 3 \end{pmatrix} + s \cdot \begin{pmatrix} -4 \\ -1 \end{pmatrix}, \, s \in \mathbb{R}$ The direction vector of line $h$ is $\begin{pmatrix} -4 \\ -1 \end{pmatrix}$, and from this, the slope $m$ can be calculated as: $m = \frac{\text{change in y}}{\text{change in x}} = \frac{-1}{-4} = \frac{1}{4}.$

3. Parallel Lines: Two lines are parallel if they have the same slope. However, in this case, line $g$ has slope $m = -\frac{1}{4}$, while the parametric equation suggests the direction vector of $h$ has slope $m = -\frac{1}{4}$.

Final Justification:

The two lines $g$ and $h$ are parallel because their slopes (derived from their equations) are equal, which is $m = -\frac{1}{4}$.

Would you like further clarification or detailed steps on how to calculate slopes or analyze line equations?

Here are five additional related questions to deepen understanding:

1. How do you calculate the slope from a parametric equation of a line?
2. What conditions must be met for two lines to be considered identical rather than just parallel?
3. How can the distance between two parallel lines be calculated?
4. What is the significance of the direction vector in parametric line equations?
5. Can we find the point of intersection if these two lines are extended infinitely?

Tip: When dealing with parametric equations, always extract the direction vector to find the slope or to compare it with other line equations.