## Problem Description and Symbolic Representation

The problem can be broken down into the following components:

- **Variables:**
  - Let \(x_1\) be the number of controllers sold.
  - Let \(x_2\) be the number of speakers sold.

- **Objective Function:**
  - The profit from selling \(x_1\) controllers is \(70x_1\).
  - The profit from selling \(x_2\) speakers is \(20x_2\).
  - Therefore, the total profit \(P\) to be maximized is \(P = 70x_1 + 20x_2\).

- **Constraints:**
  1. The store spends at most $50,000 on controllers and speakers each month.
     - The cost of \(x_1\) controllers is \(150x_1\).
     - The cost of \(x_2\) speakers is \(100x_2\).
     - Therefore, the cost constraint is \(150x_1 + 100x_2 \leq 50000\).
  
  2. A minimum of 15 but at most 60 controllers are sold each month.
     - This translates to \(15 \leq x_1 \leq 60\).

  3. The number of speakers sold is at most four times the number of controllers sold.
     - This gives \(x_2 \leq 4x_1\).

## Symbolic Representation

```json
{
  'sym_variables': [('x1', 'controllers'), ('x2', 'speakers')],
  'objective_function': 'Maximize 70*x1 + 20*x2',
  'constraints': [
    '150*x1 + 100*x2 <= 50000',
    'x1 >= 15',
    'x1 <= 60',
    'x2 <= 4*x1',
    'x1, x2 >= 0' # Assuming non-negativity
  ]
}
```

## Gurobi Code

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(lb=0, name="controllers") # Non-negativity assumed
    x2 = model.addVar(lb=0, name="speakers")   # Non-negativity assumed

    # Objective function
    model.setObjective(70*x1 + 20*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(150*x1 + 100*x2 <= 50000, name="budget_constraint")
    model.addConstr(x1 >= 15, name="min_controllers")
    model.addConstr(x1 <= 60, name="max_controllers")
    model.addConstr(x2 <= 4*x1, name="speakers_vs_controllers")

    # Update model
    model.update()

    # Solve model
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Controllers: {x1.varValue}")
        print(f"Speakers: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```