To find the probability that neither of the two balls picked is blue, we need to first calculate the total number of ways to pick two balls from the box and then calculate the number of ways to pick two balls that are not blue.

1. Total number of ways to pick two balls:
The total number of balls in the box is 6 blue + 5 green + 4 red = 15 balls. We need to calculate the number of ways to choose 2 balls from these 15. This can be done using the combination formula:
nCr = n! / (r!(n-r)!),
where n is the total number of items and r is the number of items to be chosen.

For our case, n = 15 and r = 2. Plugging these values into the formula, we get:
15C2 = 15! / (2!(15-2)!) = 15! / (2!13!) = (15*14) / (2*1) = 105.

So, there are a total of 105 ways to pick two balls from the box.

2. Number of ways to pick two balls that are not blue:
We need to calculate the number of ways to choose 2 balls from the 5 green and 4 red balls, without picking any blue balls.

Using the same combination formula, n = 9 (5 green + 4 red) and r = 2, we get:
9C2 = 9! / (2!(9-2)!) = 9! / (2!7!) = (9*8) / (2*1) = 36.

So, there are a total of 36 ways to pick two balls that are not blue.

3. Calculating the probability:
The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes.

In this case, the number of favorable outcomes is 36 (number of ways to pick two balls that are not blue) and the total number of possible outcomes is 105 (total number of ways to pick two balls).

Therefore, the probability that neither of the two balls picked is blue is:
36/105 = 12/35.

Hence, the correct answer is option B) 12/35.