To solve the expression $\left(-\frac{4}{5}\right)^3 \times \left(\frac{5}{4}\right)^2,$ we can break it down step by step.

1. Calculate each power:

   • $\left(-\frac{4}{5}\right)^3$ means we need to cube $-\frac{4}{5}$: $\left(-\frac{4}{5}\right)^3 = -\frac{4^3}{5^3} = -\frac{64}{125}$
   • $\left(\frac{5}{4}\right)^2$ means we need to square $\frac{5}{4}$: $\left(\frac{5}{4}\right)^2 = \frac{5^2}{4^2} = \frac{25}{16}$

2. Multiply the results: $-\frac{64}{125} \times \frac{25}{16} = -\frac{64 \times 25}{125 \times 16}$

3. Simplify the expression:

   • Multiply the numerators: $64 \times 25 = 1600$
   • Multiply the denominators: $125 \times 16 = 2000$
   • So, the expression becomes: $-\frac{1600}{2000}$

4. Simplify the fraction: $-\frac{1600}{2000} = -\frac{16}{20} = -\frac{4}{5}$

Thus, the final answer is $\boxed{-\frac{4}{5}}$.

Would you like any further details on this problem, or have any other questions? Here are some related questions you might find interesting:

1. How do you handle negative exponents in fractions?
2. What is the difference between multiplying and adding fractions?
3. How do you simplify complex fractions?
4. Can you explain how to deal with fractional exponents?
5. What are the steps to converting a mixed fraction to an improper fraction?

Tip: When dealing with powers of fractions, it helps to first simplify each fraction before applying the exponent.