## Step 1: Define the symbolic representation of the variables
The variables are 'intrusion analysts' and 'automatic alerts', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $4x_1^2 + 1x_1x_2 + 6x_2^2 + 2x_1 + 7x_2$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $9x_1 + 14x_2 \leq 37$ (network latency impact)
- $12x_1 + 6x_2 \leq 56$ (available bandwidth impact)
- $x_1^2 + x_2^2 \geq 14$ (network latency from intrusion analysts and automatic alerts)
- $x_1 + x_2 \geq 9$ (available bandwidth usage)
- $-2x_1 + 10x_2 \geq 0$ (relationship between intrusion analysts and automatic alerts)
- $9x_1 + 14x_2 \leq 31$ (network latency from intrusion analysts and automatic alerts, duplicate but with different upper bound)
- $x_1^2 + x_2^2 \leq 41$ (maximum available bandwidth from intrusion analysts and automatic alerts)
- $x_1, x_2 \geq 0$ and are integers.

## 4: Correct and consolidate constraints for clarity and feasibility
Upon review, we notice:
- The network latency constraint has two forms but essentially should be considered as $9x_1 + 14x_2 \leq 37$ for a direct limit.
- The correct interpretation should focus on provided limits and logical deductions.

## 5: Symbolic Representation
```json
{
'sym_variables': [('x1', 'intrusion analysts'), ('x2', 'automatic alerts')],
'objective_function': '4*x1^2 + x1*x2 + 6*x2^2 + 2*x1 + 7*x2',
'constraints': [
    '9*x1 + 14*x2 <= 37',
    '12*x1 + 6*x2 <= 56',
    'x1^2 + x2^2 >= 14',
    'x1 + x2 >= 9',
    '-2*x1 + 10*x2 >= 0',
    '9*x1 + 14*x2 <= 31',
    'x1^2 + x2^2 <= 41',
    'x1, x2 are integers'
]
}
```

## 6: Gurobi Code Implementation
```python
import gurobipy as gp

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

# Define variables
x1 = m.addVar(name="intrusion_analysts", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="automatic_alerts", vtype=gp.GRB.INTEGER)

# Objective function
m.setObjective(4*x1**2 + x1*x2 + 6*x2**2 + 2*x1 + 7*x2, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(9*x1 + 14*x2 <= 37, name="network_latency")
m.addConstr(12*x1 + 6*x2 <= 56, name="available_bandwidth")
m.addConstr(x1**2 + x2**2 >= 14, name="network_latency_squared")
m.addConstr(x1 + x2 >= 9, name="bandwidth_usage")
m.addConstr(-2*x1 + 10*x2 >= 0, name="analysts_alerts_relationship")
m.addConstr(9*x1 + 14*x2 <= 31, name="network_latency_limit")
m.addConstr(x1**2 + x2**2 <= 41, name="bandwidth_squared_limit")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Intrusion Analysts: {x1.varValue}")
    print(f"Automatic Alerts: {x2.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```