Final answer:
The equation of the hyperbola with directrices at x=2 and foci at (4,0) and (-4,0) is (x + 4)^2 - y^2 = 16.
Step-by-step explanation:
The equation of a hyperbola with directrices at x=2 and foci at (4,0) and (-4,0) can be written in the form (x ± a)^2 / b^2 - (y ± c)^2 / d^2 = 1, where a is the distance between the center of the hyperbola and the vertices, b is the distance between the center and the endpoints of the conjugate axis, and c is the distance between the center and the foci.
Given that the directrices are at x=2, the distance between the center and the vertices is 2, and the foci are at (4,0) and (-4,0), we can see that the center of the hyperbola is at the origin (0,0). Therefore, the equation of the hyperbola is x^2 / 4 - y^2 / a^2 = 1 (where a^2 - b^2 = 1).
We can see that the answer that matches this equation is option B: (x + 4)^2 - y^2 = 16.