To multiply the fractions \( \frac{4}{5} \) and \( \frac{10}{7} \), you can multiply the numerators together and the denominators together.
\[ \frac{4}{5} \times \frac{10}{7} = \frac{4 \times 10}{5 \times 7} = \frac{40}{35} \]
Next, simplify the fraction \( \frac{40}{35} \). Both the numerator and the denominator can be divided by 5:
\[ \frac{40 \div 5}{35 \div 5} = \frac{8}{7} \]
So, the final answer is:
\[ \frac{8}{7} \quad \text{or} \quad 1 \frac{1}{7} \text{ as a mixed number.} \]