## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Jean' and 'hours worked by Mary', which can be represented symbolically as $x_1$ and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $8.91x_1x_2 + 3.63x_1 + 2.92x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $14x_1 + 14x_2 \geq 39$ (total combined dollar cost per hour)
- $11x_1 + 7x_2 \geq 61$ (total combined likelihood to quit index)
- $10x_1^2 - 2x_2^2 \geq 0$ (quadratic constraint)
- $x_1^2 + x_2^2 \leq 87$ (total combined dollar cost per hour squared)
- $11x_1 + 7x_2 \leq 77$ (total combined likelihood to quit index upper bound)
- $x_1, x_2 \geq 0$ (non-negativity constraint, implicit in the problem statement but not explicitly listed)

## 4: Correct and Simplify Constraints
Upon review, notice that some constraints are repeated or similarly stated:
- The dollar cost per hour for Jean and Mary is given but used in constraints.
- The likelihood to quit index for Jean and Mary is given but used in constraints.

## 5: Formulate the problem in Gurobi
To formulate this problem in Gurobi, we need to translate the symbolic representation into Python code.

## 6: Write the Gurobi Code
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="hours_worked_by_Jean", lb=0)  # hours worked by Jean
x2 = m.addVar(name="hours_worked_by_Mary", lb=0)  # hours worked by Mary

# Define the objective function
m.setObjective(8.91 * x1 * x2 + 3.63 * x1 + 2.92 * x2, gp.GRB.MINIMIZE)

# Define the constraints
m.addConstr(14 * x1 + 14 * x2 >= 39, name="dollar_cost_constraint")
m.addConstr(11 * x1 + 7 * x2 >= 61, name="likelihood_to_quit_constraint")
m.addConstr(10 * x1**2 - 2 * x2**2 >= 0, name="quadratic_constraint")
m.addConstr(x1**2 + x2**2 <= 87, name="dollar_cost_squared_constraint")
m.addConstr(11 * x1 + 7 * x2 <= 77, name="likelihood_to_quit_upper_constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Jean: {x1.varValue}")
    print(f"Hours worked by Mary: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 7: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'hours worked by Jean'), ('x2', 'hours worked by Mary')],
    'objective_function': '8.91*x1*x2 + 3.63*x1 + 2.92*x2',
    'constraints': [
        '14*x1 + 14*x2 >= 39',
        '11*x1 + 7*x2 >= 61',
        '10*x1^2 - 2*x2^2 >= 0',
        'x1^2 + x2^2 <= 87',
        '11*x1 + 7*x2 <= 77'
    ]
}
```