## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. The variables are:
- $x_0$: light infantry companies
- $x_1$: signal platoons
- $x_2$: logistics companies
- $x_3$: medical platoons

The objective function to maximize is:
\[ 2x_0^2 + 6x_0x_1 + 3x_0x_2 + 4x_1^2 + 8x_1x_3 + 6x_2^2 + 8x_2x_3 + 4x_1 + 6x_2 + 9x_3 \]

## Step 2: List the constraints
The constraints are as follows:
1. $13x_0 \leq 136$
2. $11x_0 \leq 176$
3. $26x_1 \leq 136$
4. $15x_1 \leq 176$
5. $3x_2 \leq 136$
6. $10x_2 \leq 176$
7. $22x_3 \leq 136$
8. $20x_3 \leq 176$
9. $x_0^2 + x_2^2 + x_3^2 \geq 30$
10. $11x_0 + 20x_3 \geq 38$
11. $10x_2 + 20x_3 \geq 31$
12. $15x_1 + 10x_2 \geq 33$
13. $13x_0 + 3x_2 \leq 113$
14. $3x_2 + 22x_3 \leq 110$
15. $13x_0 + 22x_3 \leq 105$
16. $13x_0 + 26x_1 + 3x_2 + 22x_3 \leq 105$
17. $11^2x_0^2 + 15^2x_1^2 \leq 161$
18. $15^2x_1^2 + 20^2x_3^2 \leq 133$
19. $15^2x_1^2 + 10^2x_2^2 \leq 92$
20. $10x_2 + 20x_3 \leq 174$
21. $11x_0 + 20x_3 \leq 61$
22. $15x_1 + 10x_2 + 20x_3 \leq 168$
23. $11^2x_0^2 + 15^2x_1^2 + 10^2x_2^2 \leq 82$
24. $11x_0 + 15x_1 + 10x_2 + 20x_3 \leq 82$

## Step 3: Convert the problem into Gurobi code
We will use Gurobi's Python API to model and solve this problem.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="light_infantry_companies", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="signal_platoons", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="logistics_companies", vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="medical_platoons", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(2*x0**2 + 6*x0*x1 + 3*x0*x2 + 4*x1**2 + 8*x1*x3 + 6*x2**2 + 8*x2*x3 + 4*x1 + 6*x2 + 9*x3, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(13*x0 <= 136)
m.addConstr(11*x0 <= 176)
m.addConstr(26*x1 <= 136)
m.addConstr(15*x1 <= 176)
m.addConstr(3*x2 <= 136)
m.addConstr(10*x2 <= 176)
m.addConstr(22*x3 <= 136)
m.addConstr(20*x3 <= 176)
m.addConstr(x0**2 + x2**2 + x3**2 >= 30)
m.addConstr(11*x0 + 20*x3 >= 38)
m.addConstr(10*x2 + 20*x3 >= 31)
m.addConstr(15*x1 + 10*x2 >= 33)
m.addConstr(13*x0 + 3*x2 <= 113)
m.addConstr(3*x2 + 22*x3 <= 110)
m.addConstr(13*x0 + 22*x3 <= 105)
m.addConstr(13*x0 + 26*x1 + 3*x2 + 22*x3 <= 105)
m.addConstr(11**2*x0**2 + 15**2*x1**2 <= 161)
m.addConstr(15**2*x1**2 + 20**2*x3**2 <= 133)
m.addConstr(15**2*x1**2 + 10**2*x2**2 <= 92)
m.addConstr(10*x2 + 20*x3 <= 174)
m.addConstr(11*x0 + 20*x3 <= 61)
m.addConstr(15*x1 + 10*x2 + 20*x3 <= 168)
m.addConstr(11**2*x0**2 + 15**2*x1**2 + 10**2*x2**2 <= 82)
m.addConstr(11*x0 + 15*x1 + 10*x2 + 20*x3 <= 82)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Light infantry companies: {x0.varValue}")
    print(f"Signal platoons: {x1.varValue}")
    print(f"Logistics companies: {x2.varValue}")
    print(f"Medical platoons: {x3.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## Step 4: Symbolic representation of the problem
The symbolic representation is as follows:
```json
{
    'sym_variables': [
        ('x0', 'light infantry companies'),
        ('x1', 'signal platoons'),
        ('x2', 'logistics companies'),
        ('x3', 'medical platoons')
    ],
    'objective_function': '2*x0^2 + 6*x0*x1 + 3*x0*x2 + 4*x1^2 + 8*x1*x3 + 6*x2^2 + 8*x2*x3 + 4*x1 + 6*x2 + 9*x3',
    'constraints': [
        '13*x0 <= 136',
        '11*x0 <= 176',
        '26*x1 <= 136',
        '15*x1 <= 176',
        '3*x2 <= 136',
        '10*x2 <= 176',
        '22*x3 <= 136',
        '20*x3 <= 176',
        'x0^2 + x2^2 + x3^2 >= 30',
        '11*x0 + 20*x3 >= 38',
        '10*x2 + 20*x3 >= 31',
        '15*x1 + 10*x2 >= 33',
        '13*x0 + 3*x2 <= 113',
        '3*x2 + 22*x3 <= 110',
        '13*x0 + 22*x3 <= 105',
        '13*x0 + 26*x1 + 3*x2 + 22*x3 <= 105',
        '121*x0^2 + 225*x1^2 <= 161',
        '225*x1^2 + 400*x3^2 <= 133',
        '225*x1^2 + 100*x2^2 <= 92',
        '10*x2 + 20*x3 <= 174',
        '11*x0 + 20*x3 <= 61',
        '15*x1 + 10*x2 + 20*x3 <= 168',
        '121*x0^2 + 225*x1^2 + 100*x2^2 <= 82',
        '11*x0 + 15*x1 + 10*x2 + 20*x3 <= 82'
    ]
}
```