To find the product of \( \frac{2}{5} \) and \( \frac{2}{9} \), you multiply the numerators together and the denominators together:
\[ \frac{2}{5} \times \frac{2}{9} = \frac{(2 \times 2)}{(5 \times 9)} = \frac{4}{45} \]
So the correct response is \( \frac{4}{45} \).