## Step 1: Define the symbolic representation of the variables
The variables are: 
- security engineers: $x_0$
- SOC operators: $x_1$
- system administrators: $x_2$
- patches per day: $x_3$
- automatic alerts: $x_4$

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is: $2x_0^2 + 3x_1^2 + 9x_1x_2 + 4x_3^2 + 6x_4^2 + x_2$

## 3: List all the constraints in symbolic notation
Constraints:
1. $5.78x_0 + 5.57x_1 + 6.87x_2 + 0.77x_3 + 7.52x_4 \leq 707$
2. $1.81x_0 + 5.76x_1 + 5.12x_2 + 1.81x_3 + 2.47x_4 \leq 229$
3. $1.81x_2 + 1.81x_3 + 2.47x_4 \geq 22$
4. $1.81x_0 + 5.12x_2 + 1.81x_3 \geq 22$
5. $x_2^2 + x_3^2 + x_4^2 \geq 44$
6. $1.81x_0 + 5.12x_2 + 1.81x_3 \geq 44$
7. $4x_0^2 - 8x_1^2 + 4x_2^2 \geq 0$
8. $5.57x_1 + 6.87x_2 \leq 170$
9. $5.78x_0 + 0.77x_3 \leq 543$
10. $5.57x_1 + 7.52x_4 \leq 184$
11. $5.57x_1 + 0.77x_3 \leq 258$
12. $5.78x_0 + 6.87x_2 \leq 545$
13. $5.78x_0 + 5.57x_1 + 6.87x_2 + 0.77x_3 + 7.52x_4 \leq 545$
14. $5.12x_2 + 1.81x_3 \leq 65$
15. $5.76x_1 + 2.47x_4 \leq 225$
16. $1.81x_0 + 1.81x_3 \leq 90$
17. $5.76x_1^2 + 5.12x_2^2 \leq 166$
18. $1.81x_0^2 + 2.47x_4^2 \leq 189$
19. $5.76x_1 + 1.81x_3 \leq 199$
20. $1.81x_0^2 + 5.76x_1^2 \leq 71$
21. $5.12x_2^2 + 1.81x_3^2 + 2.47x_4^2 \leq 163$
22. $1.81x_0^2 + 5.12x_2^2 + 2.47x_4^2 \leq 102$
23. $1.81x_0 + 5.76x_1 + 5.12x_2 + 1.81x_3 + 2.47x_4 \leq 102$
24. $x_0, x_1, x_2, x_3, x_4 \geq 0$ and are integers.

## 4: Write down the symbolic problem representation
```json
{
    'sym_variables': [
        ('x0', 'security engineers'),
        ('x1', 'SOC operators'),
        ('x2', 'system administrators'),
        ('x3', 'patches per day'),
        ('x4', 'automatic alerts')
    ],
    'objective_function': '2*x0^2 + 3*x1^2 + 9*x1*x2 + 4*x3^2 + 6*x4^2 + x2',
    'constraints': [
        '5.78*x0 + 5.57*x1 + 6.87*x2 + 0.77*x3 + 7.52*x4 <= 707',
        '1.81*x0 + 5.76*x1 + 5.12*x2 + 1.81*x3 + 2.47*x4 <= 229',
        '1.81*x2 + 1.81*x3 + 2.47*x4 >= 22',
        '1.81*x0 + 5.12*x2 + 1.81*x3 >= 22',
        'x2^2 + x3^2 + x4^2 >= 44',
        '1.81*x0 + 5.12*x2 + 1.81*x3 >= 44',
        '4*x0^2 - 8*x1^2 + 4*x2^2 >= 0',
        '5.57*x1 + 6.87*x2 <= 170',
        '5.78*x0 + 0.77*x3 <= 543',
        '5.57*x1 + 7.52*x4 <= 184',
        '5.57*x1 + 0.77*x3 <= 258',
        '5.78*x0 + 6.87*x2 <= 545',
        '5.78*x0 + 5.57*x1 + 6.87*x2 + 0.77*x3 + 7.52*x4 <= 545',
        '5.12*x2 + 1.81*x3 <= 65',
        '5.76*x1 + 2.47*x4 <= 225',
        '1.81*x0 + 1.81*x3 <= 90',
        '5.76*x1^2 + 5.12*x2^2 <= 166',
        '1.81*x0^2 + 2.47*x4^2 <= 189',
        '5.76*x1 + 1.81*x3 <= 199',
        '1.81*x0^2 + 5.76*x1^2 <= 71',
        '5.12*x2^2 + 1.81*x3^2 + 2.47*x4^2 <= 163',
        '1.81*x0^2 + 5.12*x2^2 + 2.47*x4^2 <= 102',
        '1.81*x0 + 5.76*x1 + 5.12*x2 + 1.81*x3 + 2.47*x4 <= 102'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="security_engineers", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="SOC_operators", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="system_administrators", vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="patches_per_day", vtype=gp.GRB.INTEGER)
x4 = m.addVar(name="automatic_alerts", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(2*x0**2 + 3*x1**2 + 9*x1*x2 + 4*x3**2 + 6*x4**2 + x2, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(5.78*x0 + 5.57*x1 + 6.87*x2 + 0.77*x3 + 7.52*x4 <= 707)
m.addConstr(1.81*x0 + 5.76*x1 + 5.12*x2 + 1.81*x3 + 2.47*x4 <= 229)
m.addConstr(1.81*x2 + 1.81*x3 + 2.47*x4 >= 22)
m.addConstr(1.81*x0 + 5.12*x2 + 1.81*x3 >= 22)
m.addConstr(x2**2 + x3**2 + x4**2 >= 44)
m.addConstr(1.81*x0 + 5.12*x2 + 1.81*x3 >= 44)
m.addConstr(4*x0**2 - 8*x1**2 + 4*x2**2 >= 0)
m.addConstr(5.57*x1 + 6.87*x2 <= 170)
m.addConstr(5.78*x0 + 0.77*x3 <= 543)
m.addConstr(5.57*x1 + 7.52*x4 <= 184)
m.addConstr(5.57*x1 + 0.77*x3 <= 258)
m.addConstr(5.78*x0 + 6.87*x2 <= 545)
m.addConstr(5.78*x0 + 5.57*x1 + 6.87*x2 + 0.77*x3 + 7.52*x4 <= 545)
m.addConstr(5.12*x2 + 1.81*x3 <= 65)
m.addConstr(5.76*x1 + 2.47*x4 <= 225)
m.addConstr(1.81*x0 + 1.81*x3 <= 90)
m.addConstr(5.76*x1**2 + 5.12*x2**2 <= 166)
m.addConstr(1.81*x0**2 + 2.47*x4**2 <= 189)
m.addConstr(5.76*x1 + 1.81*x3 <= 199)
m.addConstr(1.81*x0**2 + 5.76*x1**2 <= 71)
m.addConstr(5.12*x2**2 + 1.81*x3**2 + 2.47*x4**2 <= 163)
m.addConstr(1.81*x0**2 + 5.12*x2**2 + 2.47*x4**2 <= 102)
m.addConstr(1.81*x0 + 5.76*x1 + 5.12*x2 + 1.81*x3 + 2.47*x4 <= 102)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Security Engineers: ", x0.varValue)
    print("SOC Operators: ", x1.varValue)
    print("System Administrators: ", x2.varValue)
    print("Patches per Day: ", x3.varValue)
    print("Automatic Alerts: ", x4.varValue)
else:
    print("The model is infeasible.")
```