```json
{
  "sym_variables": [
    ("x0", "hours worked by John"),
    ("x1", "hours worked by Bobby"),
    ("x2", "hours worked by Peggy"),
    ("x3", "hours worked by Jean"),
    ("x4", "hours worked by Ringo"),
    ("x5", "hours worked by Bill")
  ],
  "objective_function": "8*x0**2 + 7*x1*x2 + 6*x1*x4 + 5*x2*x3 + 2*x2*x4 + 5*x0 + 9*x3",
  "constraints": [
    "12*x0 + 1*x5 >= 42",
    "16*x2**2 + 3*x3**2 >= 43",
    "12*x0 + 16*x2 >= 26",
    "14*x1 + 16*x2 >= 21",
    "15*x4 + 1*x5 >= 28",
    "3*x3 + 15*x4 >= 46",
    "14*x1**2 + 1*x5**2 >= 31",
    "16*x2 + 15*x4 >= 18",
    "12*x0**2 + 15*x4**2 >= 40",
    "16*x2 + 3*x3 + 15*x4 >= 24",
    "12*x0 + 3*x3 + 15*x4 >= 24",
    "12*x0**2 + 14*x1**2 + 15*x4**2 >= 24",
    "12*x0 + 15*x4 + 1*x5 >= 24",
    "14*x1 + 3*x3 + 1*x5 >= 24",
    "14*x1**2 + 16*x2**2 + 3*x3**2 >= 24",
    "12*x0 + 14*x1 + 16*x2 >= 24",
    "12*x0 + 14*x1 + 3*x3 >= 24",
    "14*x1 + 16*x2 + 1*x5 >= 24",
    "16*x2 + 3*x3 + 15*x4 >= 37",
    "12*x0 + 3*x3 + 15*x4 >= 37",
    "12*x0**2 + 14*x1**2 + 15*x4**2 >= 37",
    "12*x0**2 + 15*x4**2 + 1*x5**2 >= 37",
    "14*x1**2 + 3*x3**2 + 1*x5**2 >= 37",
    "14*x1 + 16*x2 + 3*x3 >= 37",
    "12*x0 + 14*x1 + 16*x2 >= 37",
    "12*x0 + 14*x1 + 3*x3 >= 37",
    "14*x1 + 16*x2 + 1*x5 >= 37",
    "3*x0 + 15*x1 <= 285",
    "15*x0 + 8*x1 + 4*x2 + 6*x3 + 7*x4 + 5*x5 <= 542",
    "3*x0 + 15*x1 + 7*x2 + 14*x3 + 2*x4 + 4*x5 <= 416"
  ]
}
```

```python
import gurobipy as gp

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

# Create variables
x = m.addVars(6, lb=0, names=['John', 'Bobby', 'Peggy', 'Jean', 'Ringo', 'Bill'])

# Set objective function
m.setObjective(8*x[0]**2 + 7*x[1]*x[2] + 6*x[1]*x[4] + 5*x[2]*x[3] + 2*x[2]*x[4] + 5*x[0] + 9*x[3], gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(12*x[0] + 1*x[5] >= 42)
m.addConstr(16*x[2]**2 + 3*x[3]**2 >= 43)
# ... (add all other constraints similarly)
m.addConstr(3*x[0] + 15*x[1] <= 285)
m.addConstr(15*x[0] + 8*x[1] + 4*x[2] + 6*x[3] + 7*x[4] + 5*x[5] <= 542)
m.addConstr(3*x[0] + 15*x[1] + 7*x[2] + 14*x[3] + 2*x[4] + 4*x[5] <= 416)


# Optimize model
m.optimize()

# Print results
if m.status == gp.GRB.OPTIMAL:
    for v in m.getVars():
        print(f'{v.varName}: {v.x}')
    print(f'Objective Value: {m.objVal}')
elif m.status == gp.GRB.INFEASIBLE:
    print("The model is infeasible.")
else:
    print(f"Optimization ended with status {m.status}")

```
