## Problem Description and Symbolic Representation

The problem can be broken down into the following components:

- **Variables:**
  - Let $x_1$ be the number of zippered hoodies.
  - Let $x_2$ be the number of pullover hoodies.

- **Objective Function:**
  - The profit per zippered hoodie is $5, and the profit per pullover hoodie is $4. Therefore, the objective function to maximize profit is $5x_1 + 4x_2$.

- **Constraints:**
  - The cost of zippered hoodies is $20 each, and pullover hoodies is $15 each, with a total budget of $450. This gives the constraint $20x_1 + 15x_2 \leq 450$.
  - The boy can sell at most 25 hoodies total. This gives the constraint $x_1 + x_2 \leq 25$.
  - Non-negativity constraints: $x_1 \geq 0$ and $x_2 \geq 0$, as the number of hoodies cannot be negative.
  - Since $x_1$ and $x_2$ represent the number of hoodies, they must also be integers.

## Symbolic Representation

```json
{
    'sym_variables': [('x1', 'zippered hoodies'), ('x2', 'pullover hoodies')],
    'objective_function': 'Maximize 5*x1 + 4*x2',
    'constraints': [
        '20*x1 + 15*x2 <= 450',
        'x1 + x2 <= 25',
        'x1 >= 0',
        'x2 >= 0',
        'x1, x2 are integers'
    ]
}
```

## Gurobi Code

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="zippered_hoodies", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="pullover_hoodies", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(5*x1 + 4*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(20*x1 + 15*x2 <= 450, name="budget_constraint")
    model.addConstr(x1 + x2 <= 25, name="total_hoodies_constraint")
    model.addConstr(x1 >= 0, name="non_negative_x1")
    model.addConstr(x2 >= 0, name="non_negative_x2")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: x1 = {x1.varValue}, x2 = {x2.varValue}")
    else:
        print("No optimal solution found")

solve_hoodie_problem()
```