To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints algebraically.

Let's define:
- $x_1$ as the number of pounds of family mix to be made.
- $x_2$ as the number of pounds of party mix to be made.

Given that:
- The family mix sells for $4.5 a pound.
- The party mix sells for $6 a pound.
- The family mix has 50% caramel popcorn and 50% butter popcorn.
- The party mix has 80% caramel popcorn and 20% butter popcorn.
- There are 35 pounds of butter popcorn available.
- There are 45 pounds of caramel popcorn available.

The objective function, which is to maximize profit, can be written as:
\[ \text{Maximize} \quad 4.5x_1 + 6x_2 \]

The constraints based on the availability of butter and caramel popcorn are:
- For butter popcorn: $0.5x_1 + 0.2x_2 \leq 35$
- For caramel popcorn: $0.5x_1 + 0.8x_2 \leq 45$
- Non-negativity constraints: $x_1, x_2 \geq 0$

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'pounds of family mix'), ('x2', 'pounds of party mix')],
    'objective_function': '4.5*x1 + 6*x2',
    'constraints': ['0.5*x1 + 0.2*x2 <= 35', '0.5*x1 + 0.8*x2 <= 45', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's write the Gurobi code in Python to solve this linear programming problem:

```python
from gurobipy import *

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

# Add variables
x1 = m.addVar(lb=0, name="family_mix")
x2 = m.addVar(lb=0, name="party_mix")

# Set the objective function
m.setObjective(4.5*x1 + 6*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(0.5*x1 + 0.2*x2 <= 35, "butter_popcorn")
m.addConstr(0.5*x1 + 0.8*x2 <= 45, "caramel_popcorn")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Family mix: {x1.x} pounds")
    print(f"Party mix: {x2.x} pounds")
    print(f"Maximum profit: ${4.5*x1.x + 6*x2.x:.2f}")
else:
    print("No optimal solution found.")
```