To solve the given optimization problem, we first need to translate the natural language description into a mathematical formulation that can be implemented using Gurobi. The objective function and constraints are described as follows:

- Objective Function: Minimize \(8.91 \times \text{hours worked by Jean} \times \text{hours worked by Mary} + 3.63 \times \text{hours worked by Jean} + 2.92 \times \text{hours worked by Mary}\)

- Constraints:
  1. Dollar cost per hour for Jean: \(14\)
  2. Likelihood to quit index for Jean: \(11\)
  3. Dollar cost per hour for Mary: \(14\)
  4. Likelihood to quit index for Mary: \(7\)
  5. Minimum combined dollar cost per hour: \(\geq 39\)
  6. Minimum combined likelihood to quit index: \(\geq 61\)
  7. Constraint on hours worked by Jean and Mary: \(10 \times (\text{hours worked by Jean})^2 - 2 \times (\text{hours worked by Mary})^2 \geq 0\)
  8. Maximum combined dollar cost per hour squared: \(\leq 87\)
  9. Maximum combined likelihood to quit index: \(\leq 77\)

Given that the problem allows for non-whole numbers of hours worked by both Jean and Mary, we can model this as a continuous optimization problem.

Here's how we translate these requirements into Gurobi code:

```python
from gurobipy import *

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

# Define variables
hours_worked_by_jean = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_jean")
hours_worked_by_mary = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_mary")

# Objective function
m.setObjective(8.91 * hours_worked_by_jean * hours_worked_by_mary + 3.63 * hours_worked_by_jean + 2.92 * hours_worked_by_mary, GRB.MINIMIZE)

# Constraints
# Minimum combined dollar cost per hour
m.addConstr(14 * hours_worked_by_jean + 14 * hours_worked_by_mary >= 39, name="min_dollar_cost")

# Minimum combined likelihood to quit index
m.addConstr(11 * hours_worked_by_jean + 7 * hours_worked_by_mary >= 61, name="min_quit_index")

# Constraint on hours worked by Jean and Mary
m.addConstr(10 * hours_worked_by_jean**2 - 2 * hours_worked_by_mary**2 >= 0, name="hours_constraint")

# Maximum combined dollar cost per hour squared
m.addConstr((14 * hours_worked_by_jean)**2 + (14 * hours_worked_by_mary)**2 <= 87, name="max_dollar_cost_squared")

# Maximum combined likelihood to quit index
m.addConstr(11 * hours_worked_by_jean + 7 * hours_worked_by_mary <= 77, name="max_quit_index")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print("Hours worked by Jean:", hours_worked_by_jean.x)
    print("Hours worked by Mary:", hours_worked_by_mary.x)
    print("Objective function value:", m.objVal)
else:
    print("No optimal solution found")
```