## Step 1: Define the variables and their attributes
We have four variables: 'squash plants', 'coleus', 'hydrangeas', and 'cherry trees'. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively. Their attributes are given as:
- $r_0$: water need with $x_0 = 15$, $x_1 = 28$, $x_2 = 12$, $x_3 = 17$, and an upper bound of 147.
- $r_1$: dollar cost with $x_0 = 27$, $x_1 = 21$, $x_2 = 29$, $x_3 = 14$, and an upper bound of 102.

## 2: Formulate the objective function
The objective function to maximize is:
\[ 4x_0^2 + 3x_0x_1 + 5x_0x_2 + 8x_0x_3 + 8x_1^2 + 9x_1x_2 + 6x_1x_3 + 3x_0 + 6x_3 \]

## 3: Define the constraints
1. $15x_0 + 28x_1 + 12x_2 + 17x_3 \leq 147$ (water need)
2. $27x_0 + 21x_1 + 29x_2 + 14x_3 \leq 102$ (dollar cost)
3. $15x_0 + 12x_2 + 17x_3 \geq 18$ 
4. $15^2x_0^2 + 28^2x_1^2 + 17^2x_3^2 \geq 18$
5. $15x_0 + 28x_1 + 12x_2 \geq 18$
6. $28x_1 + 12x_2 + 17x_3 \geq 18$
7. $15x_0 + 12x_2 + 17x_3 \geq 34$
8. $15x_0 + 28x_1 + 17x_3 \geq 34$
9. $15x_0 + 28x_1 + 12x_2 \geq 34$
10. $28x_1 + 12x_2 + 17x_3 \geq 34$
11. $15^2x_0^2 + 12^2x_2^2 + 17^2x_3^2 \geq 19$
12. $15x_0 + 28x_1 + 17x_3 \geq 19$
13. $15x_0 + 28x_1 + 12x_2 \geq 19$
14. $28x_1 + 12x_2 + 17x_3 \geq 19$
15. $15x_0 + 12x_2 + 17x_3 \geq 34$
16. $15x_0 + 28x_1 + 17x_3 \geq 34$
17. $15x_0 + 28x_1 + 12x_2 \geq 34$
18. $28^2x_1^2 + 12^2x_2^2 + 17^2x_3^2 \geq 34$
19. $21x_1 + 29x_2 + 14x_3 \geq 19$
20. $4x_0^2 - 3x_1^2 - x_2^2 \geq 0$
21. $12^2x_2^2 + 17^2x_3^2 \leq 78$
22. $28x_1 + 17x_3 \leq 94$
23. $15^2x_0^2 + 12^2x_2^2 \leq 111$
24. $15x_0 + 28x_1 + 12x_2 + 17x_3 \leq 111$
25. $21x_1 + 14x_3 \leq 47$
26. $27x_0 + 14x_3 \leq 65$
27. $27x_0 + 29x_2 + 14x_3 \leq 65$
28. $27x_0 + 21x_1 + 29x_2 \leq 78$
29. $27^2x_0^2 + 21^2x_1^2 + 14^2x_3^2 \leq 87$
30. $27x_0 + 21x_1 + 29x_2 + 14x_3 \leq 87$
31. $x_0, x_1, x_2, x_3$ are integers.

## 4: Implement the problem in Gurobi
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()
    
    # Define variables
    x0 = model.addVar(name="squash_plants", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="coleus", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="hydrangeas", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="cherry_trees", vtype=gurobi.GRB.INTEGER)
    
    # Objective function
    model.setObjective(4*x0**2 + 3*x0*x1 + 5*x0*x2 + 8*x0*x3 + 8*x1**2 + 9*x1*x2 + 6*x1*x3 + 3*x0 + 6*x3, gurobi.GRB.MAXIMIZE)
    
    # Constraints
    model.addConstr(15*x0 + 28*x1 + 12*x2 + 17*x3 <= 147)
    model.addConstr(27*x0 + 21*x1 + 29*x2 + 14*x3 <= 102)
    model.addConstr(15*x0 + 12*x2 + 17*x3 >= 18)
    model.addConstr(15**2*x0**2 + 28**2*x1**2 + 17**2*x3**2 >= 18)
    model.addConstr(15*x0 + 28*x1 + 12*x2 >= 18)
    model.addConstr(28*x1 + 12*x2 + 17*x3 >= 18)
    model.addConstr(15*x0 + 12*x2 + 17*x3 >= 34)
    model.addConstr(15*x0 + 28*x1 + 17*x3 >= 34)
    model.addConstr(15*x0 + 28*x1 + 12*x2 >= 34)
    model.addConstr(28*x1 + 12*x2 + 17*x3 >= 34)
    model.addConstr(15**2*x0**2 + 12**2*x2**2 + 17**2*x3**2 >= 19)
    model.addConstr(15*x0 + 28*x1 + 17*x3 >= 19)
    model.addConstr(15*x0 + 28*x1 + 12*x2 >= 19)
    model.addConstr(28*x1 + 12*x2 + 17*x3 >= 19)
    model.addConstr(15*x0 + 12*x2 + 17*x3 >= 34)
    model.addConstr(15*x0 + 28*x1 + 17*x3 >= 34)
    model.addConstr(15*x0 + 28*x1 + 12*x2 >= 34)
    model.addConstr(28**2*x1**2 + 12**2*x2**2 + 17**2*x3**2 >= 34)
    model.addConstr(21*x1 + 29*x2 + 14*x3 >= 19)
    model.addConstr(4*x0**2 - 3*x1**2 - x2**2 >= 0)
    model.addConstr(12**2*x2**2 + 17**2*x3**2 <= 78)
    model.addConstr(28*x1 + 17*x3 <= 94)
    model.addConstr(15**2*x0**2 + 12**2*x2**2 <= 111)
    model.addConstr(15*x0 + 28*x1 + 12*x2 + 17*x3 <= 111)
    model.addConstr(21*x1 + 14*x3 <= 47)
    model.addConstr(27*x0 + 14*x3 <= 65)
    model.addConstr(27*x0 + 29*x2 + 14*x3 <= 65)
    model.addConstr(27*x0 + 21*x1 + 29*x2 <= 78)
    model.addConstr(27**2*x0**2 + 21**2*x1**2 + 14**2*x3**2 <= 87)
    model.addConstr(27*x0 + 21*x1 + 29*x2 + 14*x3 <= 87)
    
    # Solve the problem
    model.optimize()
    
    # Output the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("squash plants: ", x0.varValue)
        print("coleus: ", x1.varValue)
        print("hydrangeas: ", x2.varValue)
        print("cherry trees: ", x3.varValue)
    else:
        print("The problem is infeasible")

optimize_problem()
```