## Step 1: Define the symbolic representation of the variables
The variables are ['squash plants', 'coleus', 'hydrangeas', 'cherry trees']. Let's denote them as $x_1, x_2, x_3, x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is: $4x_1^2 + 3x_1x_2 + 5x_1x_3 + 8x_1x_4 + 8x_2^2 + 9x_2x_3 + 6x_2x_4 + 3x_1 + 6x_4$.

## Step 3: List the constraints in symbolic notation
1. $15x_1 \leq 147$
2. $27x_1 \leq 102$
3. $28x_2 \leq 147$
4. $21x_2 \leq 102$
5. $12x_3 \leq 147$
6. $29x_3 \leq 102$
7. $17x_4 \leq 147$
8. $14x_4 \leq 102$
9. $15x_1 + 12x_3 + 17x_4 \geq 18$
10. $15^2x_1^2 + 28^2x_2^2 + 17^2x_4^2 \geq 18$
11. $15x_1 + 28x_2 + 12x_3 \geq 18$
12. $28x_2 + 12x_3 + 17x_4 \geq 18$
13. $15x_1 + 12x_3 + 17x_4 \geq 34$
14. $15x_1 + 28x_2 + 17x_4 \geq 34$
15. $15x_1 + 28x_2 + 12x_3 \geq 34$
16. $28x_2 + 12x_3 + 17x_4 \geq 34$
17. $15^2x_1^2 + 12^2x_3^2 + 17^2x_4^2 \geq 19$
18. $15x_1 + 28x_2 + 17x_4 \geq 19$
19. $15x_1 + 28x_2 + 12x_3 \geq 19$
20. $28x_2 + 12x_3 + 17x_4 \geq 19$
21. $15x_1 + 12x_3 + 17x_4 \geq 34$
22. $15x_1 + 28x_2 + 17x_4 \geq 34$
23. $15x_1 + 28x_2 + 12x_3 \geq 34$
24. $28^2x_2^2 + 12^2x_3^2 + 17^2x_4^2 \geq 34$
25. $21x_2 + 29x_3 + 14x_4 \geq 19$
26. $4x_1^2 - 3x_2^2 - x_3^2 \geq 0$
27. $12^2x_3^2 + 17^2x_4^2 \leq 78$
28. $28x_2 + 17x_4 \leq 94$
29. $15^2x_1^2 + 12^2x_3^2 \leq 111$
30. $15x_1 + 28x_2 + 12x_3 + 17x_4 \leq 111$
31. $21x_2 + 14x_4 \leq 47$
32. $27x_1 + 14x_4 \leq 65$
33. $27x_1 + 29x_3 + 14x_4 \leq 65$
34. $27x_1 + 21x_2 + 29x_3 \leq 78$
35. $27^2x_1^2 + 21^2x_2^2 + 17^2x_4^2 \leq 87$
36. $27x_1 + 21x_2 + 29x_3 + 14x_4 \leq 87$

## Step 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'squash plants'), 
        ('x2', 'coleus'), 
        ('x3', 'hydrangeas'), 
        ('x4', 'cherry trees')
    ], 
    'objective_function': '4*x1^2 + 3*x1*x2 + 5*x1*x3 + 8*x1*x4 + 8*x2^2 + 9*x2*x3 + 6*x2*x4 + 3*x1 + 6*x4', 
    'constraints': [
        '15*x1 <= 147',
        '27*x1 <= 102',
        '28*x2 <= 147',
        '21*x2 <= 102',
        '12*x3 <= 147',
        '29*x3 <= 102',
        '17*x4 <= 147',
        '14*x4 <= 102',
        '15*x1 + 12*x3 + 17*x4 >= 18',
        '15^2*x1^2 + 28^2*x2^2 + 17^2*x4^2 >= 18',
        '15*x1 + 28*x2 + 12*x3 >= 18',
        '28*x2 + 12*x3 + 17*x4 >= 18',
        '15*x1 + 12*x3 + 17*x4 >= 34',
        '15*x1 + 28*x2 + 17*x4 >= 34',
        '15*x1 + 28*x2 + 12*x3 >= 34',
        '28*x2 + 12*x3 + 17*x4 >= 34',
        '15^2*x1^2 + 12^2*x3^2 + 17^2*x4^2 >= 19',
        '15*x1 + 28*x2 + 17*x4 >= 19',
        '15*x1 + 28*x2 + 12*x3 >= 19',
        '28*x2 + 12*x3 + 17*x4 >= 19',
        '15*x1 + 12*x3 + 17*x4 >= 34',
        '15*x1 + 28*x2 + 17*x4 >= 34',
        '15*x1 + 28*x2 + 12*x3 >= 34',
        '28*x2 + 12*x3 + 17*x4 >= 34',
        '28^2*x2^2 + 12^2*x3^2 + 17^2*x4^2 >= 34',
        '21*x2 + 29*x3 + 14*x4 >= 19',
        '4*x1^2 - 3*x2^2 - x3^2 >= 0',
        '12^2*x3^2 + 17^2*x4^2 <= 78',
        '28*x2 + 17*x4 <= 94',
        '15^2*x1^2 + 12^2*x3^2 <= 111',
        '15*x1 + 28*x2 + 12*x3 + 17*x4 <= 111',
        '21*x2 + 14*x4 <= 47',
        '27*x1 + 14*x4 <= 65',
        '27*x1 + 29*x3 + 14*x4 <= 65',
        '27*x1 + 21*x2 + 29*x3 <= 78',
        '27^2*x1^2 + 21^2*x2^2 + 17^2*x4^2 <= 87',
        '27*x1 + 21*x2 + 29*x3 + 14*x4 <= 87'
    ]
}
```

## Step 5: Implement the optimization problem using Gurobi
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name='squash_plants', vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name='coleus', vtype=gurobi.GRB.INTEGER)
x3 = m.addVar(name='hydrangeas', vtype=gurobi.GRB.INTEGER)
x4 = m.addVar(name='cherry_trees', vtype=gurobi.GRB.INTEGER)

# Define the objective function
m.setObjective(4*x1**2 + 3*x1*x2 + 5*x1*x3 + 8*x1*x4 + 8*x2**2 + 9*x2*x3 + 6*x2*x4 + 3*x1 + 6*x4, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(15*x1 <= 147)
m.addConstr(27*x1 <= 102)
m.addConstr(28*x2 <= 147)
m.addConstr(21*x2 <= 102)
m.addConstr(12*x3 <= 147)
m.addConstr(29*x3 <= 102)
m.addConstr(17*x4 <= 147)
m.addConstr(14*x4 <= 102)
m.addConstr(15*x1 + 12*x3 + 17*x4 >= 18)
m.addConstr(15**2*x1**2 + 28**2*x2**2 + 17**2*x4**2 >= 18)
m.addConstr(15*x1 + 28*x2 + 12*x3 >= 18)
m.addConstr(28*x2 + 12*x3 + 17*x4 >= 18)
m.addConstr(15*x1 + 12*x3 + 17*x4 >= 34)
m.addConstr(15*x1 + 28*x2 + 17*x4 >= 34)
m.addConstr(15*x1 + 28*x2 + 12*x3 >= 34)
m.addConstr(28*x2 + 12*x3 + 17*x4 >= 34)
m.addConstr(15**2*x1**2 + 12**2*x3**2 + 17**2*x4**2 >= 19)
m.addConstr(15*x1 + 28*x2 + 17*x4 >= 19)
m.addConstr(15*x1 + 28*x2 + 12*x3 >= 19)
m.addConstr(28*x2 + 12*x3 + 17*x4 >= 19)
m.addConstr(15*x1 + 12*x3 + 17*x4 >= 34)
m.addConstr(15*x1 + 28*x2 + 17*x4 >= 34)
m.addConstr(15*x1 + 28*x2 + 12*x3 >= 34)
m.addConstr(28*x2 + 12*x3 + 17*x4 >= 34)
m.addConstr(28**2*x2**2 + 12**2*x3**2 + 17**2*x4**2 >= 34)
m.addConstr(21*x2 + 29*x3 + 14*x4 >= 19)
m.addConstr(4*x1**2 - 3*x2**2 - x3**2 >= 0)
m.addConstr(12**2*x3**2 + 17**2*x4**2 <= 78)
m.addConstr(28*x2 + 17*x4 <= 94)
m.addConstr(15**2*x1**2 + 12**2*x3**2 <= 111)
m.addConstr(15*x1 + 28*x2 + 12*x3 + 17*x4 <= 111)
m.addConstr(21*x2 + 14*x4 <= 47)
m.addConstr(27*x1 + 14*x4 <= 65)
m.addConstr(27*x1 + 29*x3 + 14*x4 <= 65)
m.addConstr(27*x1 + 21*x2 + 29*x3 <= 78)
m.addConstr(27**2*x1**2 + 21**2*x2**2 + 17**2*x4**2 <= 87)
m.addConstr(27*x1 + 21*x2 + 29*x3 + 14*x4 <= 87)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objVal)
    print('x1: ', x1.varValue)
    print('x2: ', x2.varValue)
    print('x3: ', x3.varValue)
    print('x4: ', x4.varValue)
else:
    print('The model is infeasible')
```