Use the properties of operations and rules to solve a multi-step mathematical problem

To solve the expression \((-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \div \left(\frac{7}{15}\right)\), we will follow the order of operations and apply the properties of signed numbers.

1. **Rearranging and simplifying the division**:
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as:
\[
(-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \times \left(\frac{15}{7}\right)
\]

2. **Grouping the factors**:
We can rearrange the order of multiplication:
\[
(-4) \times \left(-\frac{3}{4}\right) \times \left(\frac{7}{5}\right) \times \left(\frac{15}{7}\right)
\]

3. **Multiplying the signed numbers**:
First, we can compute \((-4) \times \left(-\frac{3}{4}\right)\):
\[
(-4) \times \left(-\frac{3}{4}\right) = 4 \times \frac{3}{4} = 3
\]

4. **Now, substitute this back into the expression**:
Our expression now simplifies to:
\[
3 \times \left(\frac{7}{5}\right) \times \left(\frac{15}{7}\right)
\]

5. **Simplifying further**:
Notice that \(\frac{7}{5} \times \frac{15}{7}\):
\[
\frac{7}{5} \times \frac{15}{7} = \frac{15}{5} = 3
\]

6. **Final multiplication**:
Now, we multiply the result by 3:
\[
3 \times 3 = 9
\]

Thus, the final answer is:
\[
\boxed{9}
\]