## Step 1: Define the optimization problem
The problem is to maximize the objective function: $3 \times (\text{reconnaissance troops})^2 + 1 \times (\text{engineer platoons})^2 + 4 \times (\text{engineer platoons})$, subject to various constraints on resources such as mobility rating, deployment weight, logistics footprint, logistical capacity, and fun factor.

## Step 2: Identify the variables and their constraints
Let $x_0$ be the number of reconnaissance troops and $x_1$ be the number of engineer platoons. The constraints are:
- $1.36x_0 + 2.16x_1 \geq 20$ (mobility rating minimum)
- $1.36x_0 + 2.16x_1 \leq 41$ (mobility rating maximum)
- $4.77x_0 + 3.38x_1 \geq 12$ (deployment weight minimum)
- $4.77x_0 + 3.38x_1 \leq 27$ (deployment weight maximum)
- $3.67x_0 + 0.07x_1 \geq 18$ (logistics footprint minimum)
- $3.67x_0 + 0.07x_1 \leq 23$ (logistics footprint maximum)
- $0.4x_0 + 5.92x_1 \leq 30$ (logistical capacity maximum)
- $(0.4x_0)^2 + (5.92x_1)^2 \geq 20$ (logistical capacity squared minimum)
- $(5.77x_0)^2 + (7.12x_1)^2 \geq 15$ (fun factor squared minimum)
- $(5.77x_0)^2 + (7.12x_1)^2 \leq 19$ (fun factor squared maximum, corrected to reflect the actual constraint given)
- $2x_0 - 6x_1 \geq 0$ (additional constraint)
- $x_0, x_1$ are integers.

## 3: Formulate the objective function
The objective function to maximize is: $3x_0^2 + x_1^2 + 4x_1$.

## 4: Implement the problem in Gurobi
We will use Gurobi's Python API to model and solve this problem.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="reconnaissance_troops", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="engineer_platoons", vtype=gp.GRB.INTEGER)

# Objective function
m.setObjective(3*x0**2 + x1**2 + 4*x1, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(1.36*x0 + 2.16*x1 >= 20, name="mobility_rating_min")
m.addConstr(1.36*x0 + 2.16*x1 <= 41, name="mobility_rating_max")
m.addConstr(4.77*x0 + 3.38*x1 >= 12, name="deployment_weight_min")
m.addConstr(4.77*x0 + 3.38*x1 <= 27, name="deployment_weight_max")
m.addConstr(3.67*x0 + 0.07*x1 >= 18, name="logistics_footprint_min")
m.addConstr(3.67*x0 + 0.07*x1 <= 23, name="logistics_footprint_max")
m.addConstr(0.4*x0 + 5.92*x1 <= 30, name="logistical_capacity_max")
m.addConstr((0.4*x0)**2 + (5.92*x1)**2 >= 20, name="logistical_capacity_squared_min")
m.addConstr((5.77*x0)**2 + (7.12*x1)**2 >= 15, name="fun_factor_squared_min")
m.addConstr((5.77*x0)**2 + (7.12*x1)**2 <= 361, name="fun_factor_squared_max") # corrected to reflect actual upper limit of 19
m.addConstr(2*x0 - 6*x1 >= 0, name="additional_constraint")

# Solve the problem
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Reconnaissance troops: {x0.varValue}")
    print(f"Engineer platoons: {x1.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```