## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

## Step 2: Identify the variables and their corresponding symbolic representations
The variables are: 
- chrysanthemums: $x_1$
- strawberry bushes: $x_2$
- decorative cabbages: $x_3$
- boxwoods: $x_4$

## Step 3: Formulate the objective function using symbolic variables
The objective function to minimize is: $8x_1^2 + 6x_1x_2 + 4x_1x_3 + 6x_2x_3 + 6x_2x_4 + 5x_3x_4 + x_4^2 + 5x_1 + 7x_2 + 6x_3 + x_4$

## 4: List all the constraints
Constraints:
1. $18x_1 + 15x_2 + 22x_3 + 21x_4 \leq 275$
2. $18x_1 + 22x_3 \geq 68 \times 144$ (converting sq ft to sq in)
3. $(18x_1)^2 + (21x_4)^2 \geq 43 \times 144^2$ (converting sq ft to sq in)
4. $15x_2 + 21x_4 \geq 54 \times 144$ (converting sq ft to sq in)
5. $18x_1 + 15x_2 \geq 46 \times 144$ (converting sq ft to sq in)
6. $(22x_3)^2 + (21x_4)^2 \geq 63 \times 144^2$ (converting sq ft to sq in)
7. $18x_1 + 22x_3 + 21x_4 \geq 57 \times 144$ (converting sq ft to sq in)
8. $15x_2 + 22x_3 + 21x_4 \geq 57 \times 144$ (converting sq ft to sq in)
9. $18x_1 + 15x_2 + 22x_3 \geq 57 \times 144$ (converting sq ft to sq in)
10. $18x_1 + 22x_3 + 21x_4 \geq 56 \times 144$ (converting sq ft to sq in)
11. $15x_2 + 22x_3 + 21x_4 \geq 56 \times 144$ (converting sq ft to sq in)
12. $18x_1 + 15x_2 + 22x_3 \geq 56 \times 144$ (converting sq ft to sq in)
13. $18x_1 + 22x_3 + 21x_4 \geq 35 \times 144$ (converting sq ft to sq in)
14. $(15x_2)^2 + (22x_3)^2 + (21x_4)^2 \geq 35 \times 144^2$ (converting sq ft to sq in)
15. $18x_1 + 15x_2 + 22x_3 \geq 35 \times 144$ (converting sq ft to sq in)
16. $18x_1 + 15x_2 + 22x_3 + 21x_4 \geq 35 \times 144$ (converting sq ft to sq in)
17. $-7x_1^2 - 4x_3^2 + 4x_4^2 \geq 0$
18. $(15x_2)^2 + (22x_3)^2 \leq 266 \times 144^2$ (converting sq ft to sq in)
19. $(22x_3)^2 + (21x_4)^2 \leq 105 \times 144^2$ (converting sq ft to sq in)
20. $(18x_1)^2 + (15x_2)^2 \leq 218 \times 144^2$ (converting sq ft to sq in)
21. $18x_1 + 15x_2 + 22x_3 \leq 85 \times 144$ (converting sq ft to sq in)
22. $(15x_2)^2 + (22x_3)^2 + (21x_4)^2 \leq 130 \times 144^2$ (converting sq ft to sq in)
23. $18x_1 + 22x_3 + 21x_4 \leq 87 \times 144$ (converting sq ft to sq in)
24. $(18x_1)^2 + (15x_2)^2 + (21x_4)^2 \leq 96 \times 144^2$ (converting sq ft to sq in)

## 5: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [
        ('x1', 'chrysanthemums'), 
        ('x2', 'strawberry bushes'), 
        ('x3', 'decorative cabbages'), 
        ('x4', 'boxwoods')
    ], 
    'objective_function': '8*x1^2 + 6*x1*x2 + 4*x1*x3 + 6*x2*x3 + 6*x2*x4 + 5*x3*x4 + x4^2 + 5*x1 + 7*x2 + 6*x3 + x4', 
    'constraints': [
        '18*x1 + 15*x2 + 22*x3 + 21*x4 <= 275',
        '18*x1 + 22*x3 >= 68*144',
        '(18*x1)^2 + (21*x4)^2 >= 43*144^2',
        '15*x2 + 21*x4 >= 54*144',
        '18*x1 + 15*x2 >= 46*144',
        '(22*x3)^2 + (21*x4)^2 >= 63*144^2',
        '18*x1 + 22*x3 + 21*x4 >= 57*144',
        '15*x2 + 22*x3 + 21*x4 >= 57*144',
        '18*x1 + 15*x2 + 22*x3 >= 57*144',
        '18*x1 + 22*x3 + 21*x4 >= 56*144',
        '15*x2 + 22*x3 + 21*x4 >= 56*144',
        '18*x1 + 15*x2 + 22*x3 >= 56*144',
        '18*x1 + 22*x3 + 21*x4 >= 35*144',
        '(15*x2)^2 + (22*x3)^2 + (21*x4)^2 >= 35*144^2',
        '18*x1 + 15*x2 + 22*x3 >= 35*144',
        '18*x1 + 15*x2 + 22*x3 + 21*x4 >= 35*144',
        '-7*x1^2 - 4*x3^2 + 4*x4^2 >= 0',
        '(15*x2)^2 + (22*x3)^2 <= 266*144^2',
        '(22*x3)^2 + (21*x4)^2 <= 105*144^2',
        '(18*x1)^2 + (15*x2)^2 <= 218*144^2',
        '18*x1 + 15*x2 + 22*x3 <= 85*144',
        '(15*x2)^2 + (22*x3)^2 + (21*x4)^2 <= 130*144^2',
        '18*x1 + 22*x3 + 21*x4 <= 87*144',
        '(18*x1)^2 + (15*x2)^2 + (21*x4)^2 <= 96*144^2'
    ]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi

# Define the model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(name="chrysanthemums", vtype=gurobi.GRB.INTEGER)
x2 = model.addVar(name="strawberry_bushes", vtype=gurobi.GRB.INTEGER)
x3 = model.addVar(name="decorative_cabbages", vtype=gurobi.GRB.INTEGER)
x4 = model.addVar(name="boxwoods", vtype=gurobi.GRB.INTEGER)

# Define the objective function
model.setObjective(8*x1**2 + 6*x1*x2 + 4*x1*x3 + 6*x2*x3 + 6*x2*x4 + 5*x3*x4 + x4**2 + 5*x1 + 7*x2 + 6*x3 + x4, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(18*x1 + 15*x2 + 22*x3 + 21*x4 <= 275)
model.addConstr(18*x1 + 22*x3 >= 68*144)
model.addConstr((18*x1)**2 + (21*x4)**2 >= 43*144**2)
model.addConstr(15*x2 + 21*x4 >= 54*144)
model.addConstr(18*x1 + 15*x2 >= 46*144)
model.addConstr((22*x3)**2 + (21*x4)**2 >= 63*144**2)
model.addConstr(18*x1 + 22*x3 + 21*x4 >= 57*144)
model.addConstr(15*x2 + 22*x3 + 21*x4 >= 57*144)
model.addConstr(18*x1 + 15*x2 + 22*x3 >= 57*144)
model.addConstr(18*x1 + 22*x3 + 21*x4 >= 56*144)
model.addConstr(15*x2 + 22*x3 + 21*x4 >= 56*144)
model.addConstr(18*x1 + 15*x2 + 22*x3 >= 56*144)
model.addConstr(18*x1 + 22*x3 + 21*x4 >= 35*144)
model.addConstr((15*x2)**2 + (22*x3)**2 + (21*x4)**2 >= 35*144**2)
model.addConstr(18*x1 + 15*x2 + 22*x3 >= 35*144)
model.addConstr(18*x1 + 15*x2 + 22*x3 + 21*x4 >= 35*144)
model.addConstr(-7*x1**2 - 4*x3**2 + 4*x4**2 >= 0)
model.addConstr((15*x2)**2 + (22*x3)**2 <= 266*144**2)
model.addConstr((22*x3)**2 + (21*x4)**2 <= 105*144**2)
model.addConstr((18*x1)**2 + (15*x2)**2 <= 218*144**2)
model.addConstr(18*x1 + 15*x2 + 22*x3 <= 85*144)
model.addConstr((15*x2)**2 + (22*x3)**2 + (21*x4)**2 <= 130*144**2)
model.addConstr(18*x1 + 22*x3 + 21*x4 <= 87*144)
model.addConstr((18*x1)**2 + (15*x2)**2 + (21*x4)**2 <= 96*144**2)

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objval)
    print("chrysanthemums: ", x1.varValue)
    print("strawberry_bushes: ", x2.varValue)
    print("decorative_cabbages: ", x3.varValue)
    print("boxwoods: ", x4.varValue)
else:
    print("The model is infeasible")
```