To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the constraints and objective function as described.

The objective function aims to minimize \(4 \times (\text{Mbps bandwidth allocated to monitoring})^2 + 4 \times (\text{Mbps bandwidth allocated to monitoring})\).

Constraints are as follows:
1. Cost of Mbps bandwidth allocated to monitoring is $17 per unit.
2. Data integrity impact of Mbps bandwidth allocated to monitoring is 17 per unit.
3. Cost of network administrators is $9 each.
4. Data integrity impact of network administrators is 17 each.
5. Total cost of Mbps bandwidth and network administrators must be at least $15.
6. Total data integrity impact from both sources must be at least 11.
7. \(4 \times (\text{Mbps bandwidth}) - 6 \times (\text{network administrators}) \geq 0\).
8. Total cost must not exceed $55.
9. The total combined data integrity impact squared from both sources must not exceed 18.
10. Both variables are restricted to non-negative integers.

Given these points, let's express the problem in a standard optimization format:
- Let \(x_0\) be the amount of Mbps bandwidth allocated to monitoring.
- Let \(x_1\) be the number of network administrators.
- The objective function is: \(\text{Minimize} \quad 4x_0^2 + 4x_0\).
- Constraints are:
    - \(17x_0 + 9x_1 \geq 15\) (Total cost constraint, adjusted for clarity and correctness).
    - \(17x_0 + 17x_1 \geq 11\) (Data integrity impact constraint).
    - \(4x_0 - 6x_1 \geq 0\) (Mixed constraint).
    - \(17x_0 + 9x_1 \leq 55\) (Total cost upper bound).
    - \((17x_0)^2 + (17x_1)^2 \leq 18\) seems to be incorrectly stated as it involves squared terms of already large numbers, potentially leading to a very small feasible region or infeasibility. This constraint may need reevaluation but will be included as per the problem statement.
    - \(x_0, x_1 \in \mathbb{Z}^+\) (Non-negativity and integer constraints).

Now, let's implement this using Gurobi:

```python
from gurobipy import *

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

# Define the variables
x0 = m.addVar(vtype=GRB.INTEGER, name="Mbps_bandwidth_allocated_to_monitoring")
x1 = m.addVar(vtype=GRB.INTEGER, name="network_administrators")

# Objective function
m.setObjective(4*x0**2 + 4*x0, GRB.MINIMIZE)

# Constraints
m.addConstr(17*x0 + 9*x1 >= 15, "Total_cost_lower_bound")
m.addConstr(17*x0 + 17*x1 >= 11, "Data_integrity_impact_lower_bound")
m.addConstr(4*x0 - 6*x1 >= 0, "Mixed_constraint")
m.addConstr(17*x0 + 9*x1 <= 55, "Total_cost_upper_bound")
m.addConstr((17*x0)**2 + (17*x1)**2 <= 18, "Data_integrity_impact_squared_upper_bound")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Mbps bandwidth allocated to monitoring: {x0.x}")
    print(f"Network administrators: {x1.x}")
else:
    print("No optimal solution found. The model may be infeasible.")
```