To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints.

Let's define:
- $x_1$ as the number of electric cars produced per day.
- $x_2$ as the number of gas cars produced per day.

The objective function aims to maximize profit. Given that the profit per electric car is $5000 and the profit per gas car is $3000, the objective function can be represented algebraically as:
\[ \text{Maximize: } 5000x_1 + 3000x_2 \]

The constraints based on the production capacities are:
1. The electric car factory can make at most 3 electric cars per day: \( x_1 \leq 3 \)
2. The gas car factory can make at most 5 gas cars per day: \( x_2 \leq 5 \)
3. The finishing touches factory can process at most 5 cars of either type per day: \( x_1 + x_2 \leq 5 \)

Additionally, the number of cars cannot be negative:
\[ x_1 \geq 0, x_2 \geq 0 \]

So, the symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of electric cars produced per day'), ('x2', 'number of gas cars produced per day')],
    'objective_function': '5000*x1 + 3000*x2',
    'constraints': ['x1 <= 3', 'x2 <= 5', 'x1 + x2 <= 5', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's write the Gurobi code in Python to solve this optimization problem:
```python
from gurobipy import *

# Create a model
m = Model("Car Production")

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="electric_cars")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="gas_cars")

# Set the objective function
m.setObjective(5000*x1 + 3000*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 <= 3, "electric_car_factory")
m.addConstr(x2 <= 5, "gas_car_factory")
m.addConstr(x1 + x2 <= 5, "finishing_touched_factory")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Electric cars: {x1.x}")
    print(f"Gas cars: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```