Let's solve each expression step by step.

### First Expression: \( \frac{6}{5} \times \left( \frac{2}{3} - \frac{3}{5} \right) \)

1. **Calculate the subtraction inside the parentheses:**
\[
\frac{2}{3} - \frac{3}{5}
\]
To subtract these fractions, we need a common denominator. The least common denominator (LCD) of 3 and 5 is 15.

Convert each fraction:
\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\]
\[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\]

Now subtract:
\[
\frac{10}{15} - \frac{9}{15} = \frac{1}{15}
\]

2. **Multiply by \( \frac{6}{5} \):**
\[
\frac{6}{5} \times \frac{1}{15} = \frac{6 \times 1}{5 \times 15} = \frac{6}{75}
\]

3. **Simplify the fraction:**
\[
\frac{6}{75} = \frac{2}{25}
\]

So, the result of the first expression is:
\[
\frac{2}{25}
\]

### Second Expression: \( \frac{1}{4} \div \frac{3}{8} \times \frac{6}{5} \)

1. **Division of fractions:**
\[
\frac{1}{4} \div \frac{3}{8}
\]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[
\frac{1}{4} \times \frac{8}{3} = \frac{1 \times 8}{4 \times 3} = \frac{8}{12}
\]

2. **Simplify the fraction:**
\[
\frac{8}{12} = \frac{2}{3}
\]

3. **Multiply by \( \frac{6}{5} \):**
\[
\frac{2}{3} \times \frac{6}{5} = \frac{2 \times 6}{3 \times 5} = \frac{12}{15}
\]

4. **Simplify the fraction:**
\[
\frac{12}{15} = \frac{4}{5}
\]

So, the result of the second expression is:
\[
\frac{4}{5}
\]

### Summary
- The result of the first expression \( \frac{6}{5} \times \left( \frac{2}{3} - \frac{3}{5} \right) \) is \( \frac{2}{25} \).
- The result of the second expression \( \frac{1}{4} \div \frac{3}{8} \times \frac{6}{5} \) is \( \frac{4}{5} \).