## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's define the variables and the objective function, as well as the constraints.

### Variables
- $x_1$ = Number of hand-bags
- $x_2$ = Number of backpacks

### Objective Function
The objective is to maximize profit. Given that the profit per hand-bag is $75 and per backpack is $60, the objective function can be written as:
\[ \text{Maximize:} \quad 75x_1 + 60x_2 \]

### Constraints
1. Sewing time constraint: Each hand-bag requires 6 minutes, and each backpack requires 7 minutes, with 400 minutes available.
\[ 6x_1 + 7x_2 \leq 400 \]
2. Painting time constraint: Each hand-bag requires 3 minutes, and each backpack requires 5 minutes, with 600 minutes available.
\[ 3x_1 + 5x_2 \leq 600 \]
3. Non-negativity constraint: The number of hand-bags and backpacks cannot be negative.
\[ x_1 \geq 0, x_2 \geq 0 \]

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'hand-bags'), ('x2', 'backpacks')],
    'objective_function': '75*x1 + 60*x2',
    'constraints': [
        '6*x1 + 7*x2 <= 400',
        '3*x1 + 5*x2 <= 600',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="hand-bags", lb=0, vtype=gurobi.GRB.CONTINUOUS)
    x2 = model.addVar(name="backpacks", lb=0, vtype=gurobi.GRB.CONTINUOUS)

    # Objective function: Maximize 75*x1 + 60*x2
    model.setObjective(75*x1 + 60*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(6*x1 + 7*x2 <= 400, name="sewing_time")
    model.addConstr(3*x1 + 5*x2 <= 600, name="painting_time")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: hand-bags = {x1.varValue}, backpacks = {x2.varValue}")
        print(f"Max Profit: ${model.objVal:.2f}")
    else:
        print("The model is infeasible.")

solve_bag_production()
```