To solve this optimization problem, we first need to translate the given natural language description into a symbolic representation. The variables are 'hours worked by Jean' and 'hours worked by Laura', which we will denote as $x_1$ and $x_2$, respectively.

### Symbolic Representation
- **Variables:** $(x_1, \text{hours worked by Jean}), (x_2, \text{hours worked by Laura})$
- **Objective Function:** Minimize $1 \cdot x_1^2 + 7 \cdot x_1 \cdot x_2 + 1 \cdot x_2^2$
- **Constraints:**
  1. $21x_1 + 24x_2 \geq 46$ (Total paperwork competence rating)
  2. $21x_1 + 24x_2 \geq 46$ (Duplicate constraint, same as above)
  3. $1x_1 + 16x_2 \geq 39$ (Total dollar cost per hour)
  4. $1x_1 + 16x_2 \geq 39$ (Duplicate constraint, same as above)
  5. $21x_1 + 28x_2 \geq 22$ (Total work quality rating)
  6. $21x_1 + 28x_2 \geq 22$ (Duplicate constraint, same as above)
  7. $8x_1^2 - 9x_2^2 \geq 0$
  8. $21^2x_1^2 + 24^2x_2^2 \leq 90$
  9. $1x_1 + 16x_2 \leq 80$ (Total dollar cost per hour)
  10. $21x_1 + 28x_2 \leq 70$ (Total work quality rating)

### JSON Representation
```json
{
    'sym_variables': [('x1', 'hours worked by Jean'), ('x2', 'hours worked by Laura')],
    'objective_function': 'Minimize 1*x1^2 + 7*x1*x2 + 1*x2^2',
    'constraints': [
        '21*x1 + 24*x2 >= 46',
        '1*x1 + 16*x2 >= 39',
        '21*x1 + 28*x2 >= 22',
        '8*x1^2 - 9*x2^2 >= 0',
        '441*x1^2 + 576*x2^2 <= 90',
        'x1 + 16*x2 <= 80',
        '21*x1 + 28*x2 <= 70'
    ]
}
```

### Gurobi Code
```python
from gurobipy import *

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

# Add variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Jean")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Laura")

# Set objective function
m.setObjective(1*x1**2 + 7*x1*x2 + 1*x2**2, GRB.MINIMIZE)

# Add constraints
m.addConstr(21*x1 + 24*x2 >= 46, "paperwork_competence_rating")
m.addConstr(1*x1 + 16*x2 >= 39, "dollar_cost_per_hour_min")
m.addConstr(21*x1 + 28*x2 >= 22, "work_quality_rating_min")
m.addConstr(8*x1**2 - 9*x2**2 >= 0, "squared_constraint")
m.addConstr(441*x1**2 + 576*x2**2 <= 90, "paperwork_competence_squared_max")
m.addConstr(x1 + 16*x2 <= 80, "dollar_cost_per_hour_max")
m.addConstr(21*x1 + 28*x2 <= 70, "work_quality_rating_max")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Jean: {x1.x}")
    print(f"Hours worked by Laura: {x2.x}")
else:
    print("No optimal solution found")
```