## Step 1: Define the variables and the objective function
Let's denote the number of monitors as \(x_1\) and the number of gaming stations as \(x_2\). The profit per monitor sold is $40, and the profit per gaming station sold is $80. Therefore, the objective function to maximize profit is \(40x_1 + 80x_2\).

## Step 2: Define the constraints
1. **Floor Space Constraint**: A monitor requires 8 sq ft of floor space, and a gaming station requires 12 sq ft. The store has 150 sq ft of floor space available. This gives us \(8x_1 + 12x_2 \leq 150\).
2. **Marketing Requirement Constraint**: At least 40% of all items in stock must be gaming stations. This translates to \(x_2 \geq 0.4(x_1 + x_2)\) or \(0.6x_2 \geq 0.4x_1\), which simplifies to \(0.4x_1 - 0.6x_2 \leq 0\).
3. **Capital Constraint**: A monitor ties up $180 in capital, and a gaming station ties up $260. The store wants to have a maximum of $4,000 worth of capital tied up at any time. This gives us \(180x_1 + 260x_2 \leq 4000\).
4. **Non-Negativity Constraints**: The number of monitors and gaming stations cannot be negative, so \(x_1 \geq 0\) and \(x_2 \geq 0\).

## 3: Symbolic Representation
The symbolic representation of the problem is:
```json
{
'sym_variables': [('x1', 'monitors'), ('x2', 'gaming stations')],
'objective_function': '40x1 + 80x2',
'constraints': [
    '8x1 + 12x2 <= 150',
    '0.4x1 - 0.6x2 <= 0',
    '180x1 + 260x2 <= 4000',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 4: Gurobi Code
Now, let's implement this linear programming problem using Gurobi in Python:
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="monitors", lb=0, vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="gaming_stations", lb=0, vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(40 * x1 + 80 * x2, gurobi.GRB.MAXIMIZE)

    # Add the constraints
    model.addConstr(8 * x1 + 12 * x2 <= 150, name="floor_space")
    model.addConstr(0.4 * x1 - 0.6 * x2 <= 0, name="marketing_requirement")
    model.addConstr(180 * x1 + 260 * x2 <= 4000, name="capital")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Monitors: {x1.varValue}")
        print(f"Gaming Stations: {x2.varValue}")
        print(f"Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```