Let's break down the given expression step by step:

\[
\left( \frac{3}{5} - \frac{3}{4} \right) \times \left( \frac{2}{6} - \frac{1}{5} \right)^2
\]

First, we need to simplify each part inside the parentheses.

### Simplify \(\frac{3}{5} - \frac{3}{4}\):

To subtract these fractions, we need a common denominator. The least common multiple of 5 and 4 is 20.

\[
\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}
\]
\[
\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}
\]

Now subtract the fractions:

\[
\frac{12}{20} - \frac{15}{20} = \frac{12 - 15}{20} = \frac{-3}{20}
\]

### Simplify \(\frac{2}{6} - \frac{1}{5}\):

Similarly, we need a common denominator. The least common multiple of 6 and 5 is 30.

\[
\frac{2}{6} = \frac{2 \times 5}{6 \times 5} = \frac{10}{30}
\]
\[
\frac{1}{5} = \frac{1 \times 6}{5 \times 6} = \frac{6}{30}
\]

Now subtract the fractions:

\[
\frac{10}{30} - \frac{6}{30} = \frac{10 - 6}{30} = \frac{4}{30} = \frac{2}{15}
\]

### Combine the results:

Now we have:

\[
\left( \frac{-3}{20} \right) \times \left( \frac{2}{15} \right)^2
\]

First, square \(\frac{2}{15}\):

\[
\left( \frac{2}{15} \right)^2 = \frac{2^2}{15^2} = \frac{4}{225}
\]

Now multiply \(\frac{-3}{20}\) by \(\frac{4}{225}\):

\[
\frac{-3}{20} \times \frac{4}{225} = \frac{-3 \times 4}{20 \times 225} = \frac{-12}{4500}
\]

Simplify \(\frac{-12}{4500}\):

\[
\frac{-12}{4500} = \frac{-12 \div 12}{4500 \div 12} = \frac{-1}{375}
\]

So, the simplified result of the given expression is:

\[
\boxed{\frac{-1}{375}}
\]