## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation.

Let's define the symbolic variables as follows:

- $x_1$ represents the number of soccer balls
- $x_2$ represents the number of basket balls

The objective is to maximize profit. The profit per soccer ball is $5, and the profit per basket ball is $8. Therefore, the objective function can be represented as:

Maximize $5x_1 + 8x_2$

The constraints based on the given resources are:

- Sewing time: $20x_1 + 15x_2 \leq 5000$
- Quality checking time: $10x_1 + 12x_2 \leq 4500$
- Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'soccer balls'), ('x2', 'basket balls')],
    'objective_function': '5*x1 + 8*x2',
    'constraints': [
        '20*x1 + 15*x2 <= 5000',
        '10*x1 + 12*x2 <= 4500',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

# Define the variables
x1 = model.addVar(name="soccer_balls", lb=0, vtype=gp.GRB.INTEGER)
x2 = model.addVar(name="basket_balls", lb=0, vtype=gp.GRB.INTEGER)

# Define the objective function
model.setObjective(5 * x1 + 8 * x2, gp.GRB.MAXIMIZE)

# Add constraints
model.addConstr(20 * x1 + 15 * x2 <= 5000, name="sewing_time")
model.addConstr(10 * x1 + 12 * x2 <= 4500, name="quality_checking_time")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution: soccer balls = {x1.varValue}, basket balls = {x2.varValue}")
    print(f"Maximum profit: ${5 * x1.varValue + 8 * x2.varValue}")
else:
    print("No optimal solution found.")
```