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

Let's define the variables as follows:
- $x_1$ represents the number of soccer balls made.
- $x_2$ represents the number of basketballs made.

The objective function is to maximize profit. Given that the profit per soccer ball is $5 and per basketball is $8, the objective function can be written as:
\[ \text{Maximize:} \quad 5x_1 + 8x_2 \]

The constraints are based on the available time for sewing and quality checking:
- For sewing: $20x_1 + 15x_2 \leq 5000$ (since each soccer ball requires 20 minutes of sewing and each basketball requires 15 minutes, and there are 5000 minutes available).
- For quality checking: $10x_1 + 12x_2 \leq 4500$ (since each soccer ball requires 10 minutes of quality checking and each basketball requires 12 minutes, and there are 4500 minutes available).

Additionally, we have non-negativity constraints because the number of balls cannot be negative:
- $x_1 \geq 0$
- $x_2 \geq 0$

Thus, the symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'number of soccer balls'), ('x2', 'number of basketballs')],
  'objective_function': '5*x1 + 8*x2',
  'constraints': ['20*x1 + 15*x2 <= 5000', '10*x1 + 12*x2 <= 4500', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's implement this problem in Gurobi using Python:

```python
from gurobipy import *

# Create a new model
m = Model("Soccer_and_Basketball_Production")

# Define the variables
x1 = m.addVar(name="soccer_balls", vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name="basketballs", vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function
m.setObjective(5*x1 + 8*x2, GRB.MAXIMIZE)

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

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: Produce {x1.x} soccer balls and {x2.x} basketballs.")
else:
    print("No optimal solution found.")
```