n is the fixed number of trials. x is the specified number of sucesses. N - X is the number of failures. P is the probablity of success on any given trail. 1 - P is the probabilty of failure on any given trial.
These probabilities hold for any value of X between 0 (lowest number of possible successes in n trials) and n (highest number of possible successes).
Because this is a fourth degree that will be tricky to factor as it is, we will do a u substitution. Let . We can now rewrite that polynomial in terms of u: . Filling into the quadratic formula we have . Simplifying down and or . That means that u = 8 or u = -1. But don't forget that we let , so we have to put x-squared back in for u. That gives us which simplifies down to . That also gives us . When we take the square root of -1, we have to use the fact that -1 = i^2, so we sub that in to get . All in all, your solutions are as follows: which is choice b.