## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'Mbps bandwidth allocated to monitoring' and 'network administrators'. Let's denote 'Mbps bandwidth allocated to monitoring' as $x_1$ and 'network administrators' as $x_2$. The objective function to minimize is $4x_1^2 + 4x_1$. The constraints are:
- $17x_1$ costs $17x_1$ dollars,
- $17x_1$ data integrity impact,
- $9x_2$ costs $9x_2$ dollars,
- $17x_2$ data integrity impact,
- $17x_1 + 9x_2 \geq 15$ (spend at least $15),
- $17x_1 + 17x_2 \geq 11$ (data integrity impact at least 11),
- $4x_1 - 6x_2 \geq 0$,
- $17x_1 + 9x_2 \leq 57$ (spend no more than $57),
- $x_1^2 + x_2^2 \leq 18$ (combined data integrity impact squared),
- $x_1$ is an integer (non-fractional Mbps bandwidth),
- $x_2$ is an integer (whole number of network administrators).

## 2: Convert the problem into a symbolic representation
The symbolic variables are:
- $x_1$ for 'Mbps bandwidth allocated to monitoring',
- $x_2$ for 'network administrators'.

The objective function is: $4x_1^2 + 4x_1$.

The constraints are:
1. $17x_1 + 9x_2 \geq 15$,
2. $17x_1 + 17x_2 \geq 11$,
3. $4x_1 - 6x_2 \geq 0$,
4. $17x_1 + 9x_2 \leq 57$,
5. $x_1^2 + x_2^2 \leq 18$,
6. $x_1$ is an integer,
7. $x_2$ is an integer.

## 3: Write down the problem in the required JSON format
```json
{
    'sym_variables': [('x1', 'Mbps bandwidth allocated to monitoring'), ('x2', 'network administrators')],
    'objective_function': '4*x1^2 + 4*x1',
    'constraints': [
        '17*x1 + 9*x2 >= 15',
        '17*x1 + 17*x2 >= 11',
        '4*x1 - 6*x2 >= 0',
        '17*x1 + 9*x2 <= 57',
        'x1^2 + x2^2 <= 18',
        'x1 is an integer',
        'x2 is an integer'
    ]
}
```

## 4: Implement the problem using Gurobi
```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(name='Mbps_bandwidth', vtype=gurobi.GRB.INTEGER)  # Mbps bandwidth allocated to monitoring
x2 = model.addVar(name='network_administrators', vtype=gurobi.GRB.INTEGER)  # network administrators

# Objective function: minimize 4*x1^2 + 4*x1
model.setObjective(4*x1**2 + 4*x1, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(17*x1 + 9*x2 >= 15, name='budget_min')
model.addConstr(17*x1 + 17*x2 >= 11, name='data_integrity_min')
model.addConstr(4*x1 - 6*x2 >= 0, name='ratio_constraint')
model.addConstr(17*x1 + 9*x2 <= 57, name='budget_max')
model.addConstr(x1**2 + x2**2 <= 18, name='data_integrity_max')

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print('Optimal solution found.')
    print(f'Mbps bandwidth allocated to monitoring: {x1.varValue}')
    print(f'network administrators: {x2.varValue}')
    print(f'Objective: {model.objVal}')
else:
    print('No optimal solution found.')
```