To solve the expression ( \frac{28}{3} : \left( \frac{5}{6} - \frac{7}{2} \right) + \left( \frac{3}{5} + 2 \frac{2}{3} \right) : \left( -\frac{98}{45} \right) ), we will break it down step by step.

Step 1: Calculate ( \frac{5}{6} - \frac{7}{2} )

First, convert ( \frac{7}{2} ) to a fraction with a denominator of 6:
[
\frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6}
]

Now perform the subtraction:
[
\frac{5}{6} - \frac{21}{6} = \frac{5 - 21}{6} = \frac{-16}{6} = \frac{-8}{3}
]

Step 2: Calculate ( \frac{28}{3} : \left( \frac{-8}{3} \right) )

Dividing ( \frac{28}{3} ) by ( \frac{-8}{3} ) is equivalent to multiplying by the reciprocal:
[
\frac{28}{3} \div \frac{-8}{3} = \frac{28}{3} \times \frac{-3}{8} = \frac{28 \times -3}{3 \times 8} = \frac{-84}{24} = \frac{-7}{2} \quad \text{(simplified)}
]

Step 3: Calculate ( 2 \frac{2}{3} )

Convert ( 2 \frac{2}{3} ) to an improper fraction:
[
2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}
]

Step 4: Calculate ( \frac{3}{5} + 2 \frac{2}{3} )

Next, add ( \frac{3}{5} + \frac{8}{3} ). To do this, find a common denominator (which is 15):
[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
]
[
\frac{8}{3} = \frac{8 \times 5}{3 \times 5} = \frac{40}{15}
]

Now add the fractions:
[
\frac{9}{15} + \frac{40}{15} = \frac{49}{15}
]

Step 5: Calculate ( \left( \frac{49}{15} \right) : \left( -\frac{98}{45} \right) )

Dividing ( \frac{49}{15} ) by ( -\frac{98}{45} ) is equivalent to multiplying by the reciprocal:
[
\frac{49}{15} \div \left( -\frac{98}{45} \right) = \frac{49}{15} \times \left( -\frac{45}{98} \right) = \frac{49 \times -45}{15 \times 98} = -\frac{2205}{1470}
]

Now simplify ( -\frac{2205}{1470} ):
[
\text{Finding GCD of 2205 and 1470, we can simplify:}
]
[
2205 = 3 \times 5 \times 7 \times 21
]
[
1470 = 2 \times 3 \times 5 \times 49
]
[
\text{Common factors: } 3 \times 5
]
Thus:
[
-\frac{2205 \div 15}{1470 \div 15} = -\frac{147}{98} = -\frac{21}{14} = -\frac{3}{2}
]

Step 6: Combine Results

Now combine results from steps 2 and 5:
[
-\frac{7}{2} + \left(-\frac{3}{2}\right) = -\frac{7 + 3}{2} = -\frac{10}{2} = -5
]

Thus, the final result is:
[
\boxed{-5}
]