```json
{
  "sym_variables": [
    ("x0", "hours worked by Bill"),
    ("x1", "hours worked by Laura"),
    ("x2", "hours worked by Dale"),
    ("x3", "hours worked by Peggy")
  ],
  "objective_function": "4*x0**2 + 9*x0*x1 + 6*x0*x2 + 1*x0*x3 + 6*x1**2 + 4*x1*x2 + 6*x1*x3 + 5*x2**2 + 2*x2*x3 + 7*x3**2 + 1*x0 + 7*x1 + 7*x2 + 4*x3",
  "constraints": [
    "19*x0 + 6*x1 >= 27",
    "6*x1**2 + 16*x3**2 >= 20",
    "19*x0 + 6*x1 + 5*x2 >= 24",
    "6*x1**2 + 5*x2**2 <= 106",
    "6*x1**2 + 16*x3**2 <= 90",
    "6*x1 + 5*x2 + 16*x3 <= 55",
    "19*x0 + 6*x1 + 16*x3 <= 159",
    "19*x0 + 6*x1 + 5*x2 + 16*x3 <= 159",
    "9*x0**2 + 3*x2**2 <= 105",
    "9*x0 + 20*x3 <= 52",
    "3*x2 + 20*x3 <= 62",
    "5*x1 + 20*x3 <= 100",
    "9*x0 + 5*x1 + 3*x2 + 20*x3 <= 100"
  ]
}
```

```python
import gurobipy as gp

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

    # Create variables
    x = model.addVars(4, lb=0, vtype=gp.GRB.CONTINUOUS, name=["Bill", "Laura", "Dale", "Peggy"])


    # Set objective function
    obj = 4*x[0]**2 + 9*x[0]*x[1] + 6*x[0]*x[2] + 1*x[0]*x[3] + 6*x[1]**2 + 4*x[1]*x[2] + 6*x[1]*x[3] + 5*x[2]**2 + 2*x[2]*x[3] + 7*x[3]**2 + 1*x[0] + 7*x[1] + 7*x[2] + 4*x[3]
    model.setObjective(obj, gp.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(19*x[0] + 6*x[1] >= 27)
    model.addConstr(6*x[1]**2 + 16*x[3]**2 >= 20)
    model.addConstr(19*x[0] + 6*x[1] + 5*x[2] >= 24)
    model.addConstr(6*x[1]**2 + 5*x[2]**2 <= 106)
    model.addConstr(6*x[1]**2 + 16*x[3]**2 <= 90)
    model.addConstr(6*x[1] + 5*x[2] + 16*x[3] <= 55)
    model.addConstr(19*x[0] + 6*x[1] + 16*x[3] <= 159)
    model.addConstr(19*x[0] + 6*x[1] + 5*x[2] + 16*x[3] <= 159)
    model.addConstr(9*x[0]**2 + 3*x[2]**2 <= 105)
    model.addConstr(9*x[0] + 20*x[3] <= 52)
    model.addConstr(3*x[2] + 20*x[3] <= 62)
    model.addConstr(5*x[1] + 20*x[3] <= 100)
    model.addConstr(9*x[0] + 5*x[1] + 3*x[2] + 20*x[3] <= 100)


    # Optimize model
    model.optimize()

    if model.status == gp.GRB.OPTIMAL:
        for v in model.getVars():
            print('%s %g' % (v.varName, v.x))
        print('Obj: %g' % model.objVal)
    elif model.status == gp.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print(f"Optimization ended with status {model.status}")


except gp.GurobiError as e:
    print('Error code ' + str(e.errno) + ': ' + str(e))

except AttributeError:
    print('Encountered an attribute error')
```