Consider the line passing through the points \( (p ; 8) \) and \( (4 ; 12) \) and which is parallel to the line \( y=2 x+5 \). The value of \( p \) is select one: \[ \text { a. } p=2 \] b. \( p=4 \) c. \( p=6 \)

To find the value of \( p \), we need to note that parallel lines have the same slope. The slope of the line given by the equation \( y = 2x + 5 \) is \( 2 \). Therefore, the line that passes through the points \( (p, 8) \) and \( (4, 12) \) must also have a slope of \( 2 \). We can calculate the slope between these two points using the slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 8}{4 - p} = \frac{4}{4 - p} \] Setting the slopes equal to each other gives us the equation: \[ \frac{4}{4 - p} = 2 \] Now, we can cross-multiply: \[ 4 = 2(4 - p) \] \[ 4 = 8 - 2p \] \[ 2p = 8 - 4 \] \[ 2p = 4 \] \[ p = 2 \] Thus, the value of \( p \) is \( \boxed{2} \).