## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the area in square feet for sunflowers.
- $x_2$ represents the area in square feet for roses.

## Step 2: Formulate the objective function
The profit per square foot of sunflowers is $450, and the profit per square foot of roses is $100. The objective is to maximize profit. Therefore, the objective function can be written as:
\[ \text{Maximize:} \quad 450x_1 + 100x_2 \]

## 3: Define the constraints
1. The total area constraint: The gardener has a field of 100 square feet.
\[ x_1 + x_2 \leq 100 \]
2. The budget constraint: The seed for sunflowers costs $67 per square foot, and the seed for roses costs $52 per square foot. The gardener has a budget of $6500.
\[ 67x_1 + 52x_2 \leq 6500 \]
3. Non-negativity constraints: The area cannot be negative.
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

## 4: Symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'sunflowers area'), ('x2', 'roses area')],
    'objective_function': '450*x1 + 100*x2',
    'constraints': [
        'x1 + x2 <= 100',
        '67*x1 + 52*x2 <= 6500',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Step 5: Gurobi code in Python
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="sunflowers_area", lb=0, ub=100)
    x2 = model.addVar(name="roses_area", lb=0, ub=100)

    # Objective function: Maximize 450*x1 + 100*x2
    model.setObjective(450 * x1 + 100 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x1 + x2 <= 100, name="total_area")
    model.addConstr(67 * x1 + 52 * x2 <= 6500, name="budget")

    # Optimize the model
    model.optimize()

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

solve_gardener_problem()
```