To find the probability that both balls taken from the bag are white, we can calculate it as follows:

1. Total balls in the bag: 4 white + 6 red = 10 balls.

2. Probability of first ball being white:
\[
\frac{4}{10} = \frac{2}{5}
\]

3. After removing one white ball, there are 3 white balls left and 9 balls in total.

Probability of second ball being white:
\[
\frac{3}{9} = \frac{1}{3}
\]

4. Now, multiply the probabilities:
\[
\frac{2}{5} \times \frac{1}{3} = \frac{2}{15}
\]

So, the probability that both balls taken are white is \(\frac{2}{15}\).

The correct answer is C. 2/15.