To simplify the given expression step-by-step, let's break it down:

\[
\frac{2}{11} \times \frac{-4}{13} + \frac{2}{11} \times \frac{-5}{13} - \frac{2}{11} \times \frac{-9}{14}
\]

First, simplify each term individually:

1. \(\frac{2}{11} \times \frac{-4}{13}\):
\[
\frac{2 \times -4}{11 \times 13} = \frac{-8}{143}
\]

2. \(\frac{2}{11} \times \frac{-5}{13}\):
\[
\frac{2 \times -5}{11 \times 13} = \frac{-10}{143}
\]

3. \(\frac{2}{11} \times \frac{-9}{14}\):
\[
\frac{2 \times -9}{11 \times 14} = \frac{-18}{154}
\]

Now, let's combine these fractions. To do this, we need a common denominator. The denominators are 143 and 154. The least common multiple (LCM) of 143 and 154 is 21979.

Convert each fraction to have the common denominator of 21979:

1. \(\frac{-8}{143}\):
\[
\frac{-8 \times 154}{143 \times 154} = \frac{-1232}{21979}
\]

2. \(\frac{-10}{143}\):
\[
\frac{-10 \times 154}{143 \times 154} = \frac{-1540}{21979}
\]

3. \(\frac{-18}{154}\):
\[
\frac{-18 \times 143}{154 \times 143} = \frac{-2574}{21979}
\]

Now, add these fractions together:
\[
\frac{-1232}{21979} + \frac{-1540}{21979} - \frac{-2574}{21979}
\]

Combine the numerators:
\[
\frac{-1232 - 1540 + 2574}{21979} = \frac{-2772 + 2574}{21979} = \frac{-198}{21979}
\]

So, the simplified form of the given expression is:
\[
\frac{-198}{21979}
\]