To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints using algebraic terms.

Let's define the variables as follows:
- $x_1$ represents the hours worked by Hank.
- $x_2$ represents the hours worked by George.

The objective function is to maximize: $1 \cdot x_1 + 9 \cdot x_2$.

Now, let's list out the constraints based on the given description:

1. The total combined likelihood to quit index from hours worked by Hank and George must be equal to or greater than 22: $4x_1 + 2x_2 \geq 22$.
2. The total combined dollar cost per hour from hours worked by Hank and George should be at least 30: $7x_1 + 2x_2 \geq 30$.
3. Three times the number of hours worked by Hank, plus -4 times the number of hours worked by George must be greater than or equal to zero: $3x_1 - 4x_2 \geq 0$.
4. The total combined likelihood to quit index from hours worked by Hank and George has to be equal to or less than 55: $4x_1 + 2x_2 \leq 55$.
5. The total combined dollar cost per hour from hours worked by Hank plus hours worked by George should be less than or equal to 51: $7x_1 + 2x_2 \leq 51$.
6. The number of hours worked by Hank must be a whole number: $x_1 \in \mathbb{Z}$.
7. There has to be an integer amount of hours worked by George: $x_2 \in \mathbb{Z}$.

Given these definitions, the symbolic representation of the problem is:

```json
{
  'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by George')],
  'objective_function': 'maximize 1*x1 + 9*x2',
  'constraints': [
    '4*x1 + 2*x2 >= 22',
    '7*x1 + 2*x2 >= 30',
    '3*x1 - 4*x2 >= 0',
    '4*x1 + 2*x2 <= 55',
    '7*x1 + 2*x2 <= 51',
    'x1 is an integer',
    'x2 is an integer'
  ]
}
```

To solve this problem using Gurobi, we will use Python. First, ensure you have Gurobi installed in your environment. If not, you can install it via the Gurobi website or through a package manager like pip for the Python interface.

Here's how you could implement this optimization problem in Gurobi:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Hank")
x2 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_George")

# Set the objective function
m.setObjective(1*x1 + 9*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(4*x1 + 2*x2 >= 22, "Likelihood_to_Quit_Index_Constraint")
m.addConstr(7*x1 + 2*x2 >= 30, "Dollar_Cost_Per_Hour_Constraint_Lower_Bound")
m.addConstr(3*x1 - 4*x2 >= 0, "Hours_Worked_Relationship_Constraint")
m.addConstr(4*x1 + 2*x2 <= 55, "Likelihood_to_Quit_Index_Upper_Bound")
m.addConstr(7*x1 + 2*x2 <= 51, "Dollar_Cost_Per_Hour_Constraint_Upper_Bound")

# Optimize the model
m.optimize()

# Print the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print("Objective:", m.objVal)
```