$4 \frac{5}{9} \times \frac{2}{5}$

Step 1 ：Convert the mixed number $4 \frac{5}{9}$ into an improper fraction. This is done by multiplying the whole number by the denominator of the fraction and adding the numerator. The result is the numerator of the improper fraction, and the denominator remains the same. So, $4 \frac{5}{9}$ becomes $\frac{4*9 + 5}{9} = \frac{36 + 5}{9} = \frac{41}{9}$.

Step 2 ：Multiply the two fractions $\frac{41}{9}$ and $\frac{2}{5}$. When multiplying fractions, we simply multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. So, $\frac{41}{9} \times \frac{2}{5} = \frac{41*2}{9*5} = \frac{82}{45}$.

Step 3 ：The fraction $\frac{82}{45}$ is already in its simplest form, so we don't need to simplify it further. So, $4 \frac{5}{9} \times \frac{2}{5} = \frac{82}{45}$.

Step 4 ：Check the result by converting the improper fraction back into a mixed number. $\frac{82}{45} = 1 \frac{37}{45}$. So, our result is correct.

Step 5 ：\(\boxed{\frac{82}{45}}\) is the final answer.