Calculate the slope (m) of the line.

The formula for the slope is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Using $P(4, 6)$ as $(x_1, y_1)$ and $S(-4, 0)$ as $(x_2, y_2)$:

$$m = \frac{0 - 6}{-4 - 4} = \frac{-6}{-8} = \frac{3}{4}$$

Using the slope $m = \frac{3}{4}$ and point $S(-4, 0)$:

$$y - y_1 = m(x - x_1)$$ $$y - 0 = \frac{3}{4}(x - (-4))$$ $$y = \frac{3}{4}(x + 4)$$ $$y = \frac{3}{4}x + \frac{3}{4} \times 4$$ $$y = \frac{3}{4}x + 3$$

This is the equation in the form $y = mx + c$.

$$\boxed{y = \frac{3}{4}x + 3}$$