To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables for each object (hydrangeas and aloe vera), formulating the objective function using these variables, and then listing all constraints in terms of these variables.

Let's define:
- $x_1$ as the number of hydrangeas,
- $x_2$ as the number of aloe vera.

The objective function to minimize is given by: 
\[1 \cdot x_1^2 + 5 \cdot x_1 + 9 \cdot x_2\]

Constraints are:
1. Resilience index from hydrangeas: $3x_1$
2. Cost of hydrangeas: $27x_1$
3. Resilience index from aloe vera: $26x_2$
4. Cost of aloe vera: $19x_2$
5. Total resilience index must be at least 52: $3x_1 + 26x_2 \geq 52$
6. Same as constraint 5, emphasizing the requirement.
7. Minimum cost of hydrangeas and aloe vera combined: $27x_1 + 19x_2 \geq 45$
8. Same as constraint 7, rephrased.
9. Additional constraint on squared terms: $-8x_1^2 + 5x_2^2 \geq 0$
10. Maximum total resilience index: $3x_1 + 26x_2 \leq 120$
11. Maximum cost constraint on squared terms of quantities: $27x_1^2 + 19x_2^2 \leq 172$
12. Hydrangeas must be a non-fractional number (integer): $x_1 \in \mathbb{Z}$
13. Aloe vera must be a whole number (integer): $x_2 \in \mathbb{Z}$

Thus, the symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'hydrangeas'), ('x2', 'aloe vera')],
  'objective_function': 'x1^2 + 5*x1 + 9*x2',
  'constraints': [
    '3*x1 + 26*x2 >= 52',
    '27*x1 + 19*x2 >= 45',
    '-8*x1^2 + 5*x2^2 >= 0',
    '3*x1 + 26*x2 <= 120',
    '27*x1^2 + 19*x2^2 <= 172',
    'x1 >= 0', 
    'x2 >= 0',
    'x1 == int(x1)', 
    'x2 == int(x2)'
  ]
}
```

To solve this problem using Gurobi, we need to write Python code that sets up the model according to these constraints and the objective function. Here's how you can do it:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="hydrangeas")
x2 = m.addVar(vtype=GRB.INTEGER, name="aloe vera")

# Set objective function
m.setObjective(x1**2 + 5*x1 + 9*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(3*x1 + 26*x2 >= 52, "Resilience_Index_Min")
m.addConstr(27*x1 + 19*x2 >= 45, "Cost_Min")
m.addConstr(-8*x1**2 + 5*x2**2 >= 0, "Squared_Terms_Constraint")
m.addConstr(3*x1 + 26*x2 <= 120, "Resilience_Index_Max")
m.addConstr(27*x1**2 + 19*x2**2 <= 172, "Cost_Squared_Terms_Max")

# Optimize model
m.optimize()

# Print results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print("Objective:", m.objVal)
```