To convert the given problem description into a symbolic representation and then solve it using Gurobi, we first need to identify the variables, objective function, and constraints based on the provided information.

### Symbolic Representation:

- **Variables:**
  - `x0`: Mbps bandwidth allocated to monitoring
  - `x1`: network administrators

- **Objective Function:**
  The objective is to minimize `4*x0^2 + 4*x0`.

- **Constraints:**
  1. Cost constraint for Mbps bandwidth allocated to monitoring: `17*x0`
  2. Data integrity impact of Mbps bandwidth allocated to monitoring: `17*x0`
  3. Cost per network administrator: `9*x1`
  4. Data integrity impact per network administrator: `17*x1`
  5. Minimum total cost constraint: `17*x0 + 9*x1 >= 15`
  6. Minimum total data integrity impact constraint: `17*x0 + 17*x1 >= 11`
  7. Combined linear constraint: `4*x0 - 6*x1 >= 0`
  8. Maximum total cost constraint: `17*x0 + 9*x1 <= 55`
  9. Maximum combined data integrity impact squared constraint: `(17*x0)^2 + (17*x1)^2 <= 18` (However, this constraint seems unusual and might be a misunderstanding since the original problem description does not clearly specify squaring the impacts for this particular constraint. Normally, such constraints would involve linear or possibly quadratic terms but not squared sums of products as it suggests an incorrectly transcribed condition.)
  10. Non-negativity and integer constraints: `x0` must be a non-negative integer, and `x1` must also be a non-negative integer.

Given the nature of the problem and to adhere strictly to the instructions provided, we'll proceed with creating a symbolic representation and Gurobi code for solving this optimization problem. Note that due to potential ambiguity in constraint 9, it will be interpreted as likely intended to reflect a quadratic relationship rather than squaring sums of products.

### Symbolic Representation in JSON Format:
```json
{
    'sym_variables': [('x0', 'Mbps bandwidth allocated to monitoring'), ('x1', 'network administrators')],
    'objective_function': '4*x0^2 + 4*x0',
    'constraints': [
        '17*x0 + 9*x1 >= 15', 
        '17*x0 + 17*x1 >= 11', 
        '4*x0 - 6*x1 >= 0', 
        '17*x0 + 9*x1 <= 55', 
        '(17*x0)^2 + (17*x1)^2 <= 18'
    ]
}
```

### Gurobi Code:
```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")

# Set the objective function
m.setObjective(4*x0**2 + 4*x0, GRB.MINIMIZE)

# Add constraints
m.addConstr(17*x0 + 9*x1 >= 15, "Minimum_total_cost")
m.addConstr(17*x0 + 17*x1 >= 11, "Minimum_total_data_integrity_impact")
m.addConstr(4*x0 - 6*x1 >= 0, "Combined_linear_constraint")
m.addConstr(17*x0 + 9*x1 <= 55, "Maximum_total_cost")
# Note: The following constraint might need adjustment based on the actual problem intent
m.addConstr((17*x0)**2 + (17*x1)**2 <= 18, "Maximum_combined_data_integrity_impact_squared")

# Solve the model
m.optimize()

# Print the solution
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print(f"Objective: {m.objVal}")
```