## Step 1: Define the symbolic representation of the variables
The variables are 'vincas', 'tomato vines', 'strawberry bushes', and 'ferns', which can be represented symbolically as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $4x_1^2 + 9x_1x_2 + 9x_1x_3 + 4x_1x_4 + 9x_2x_3 + 6x_3^2 + 6x_3x_4 + 4x_4^2 + 2x_2$.

## Step 3: List the constraints in symbolic notation
1. $2x_1 \leq 92$
2. $x_2 \leq 92$
3. $10x_3 \leq 92$
4. $9x_4 \leq 92$
5. $2x_1 + 9x_4 \geq 12$
6. $x_1^2 + x_3^2 \geq 22$
7. $x_2 + 10x_3 \geq 9$
8. $2x_1 + x_2 \geq 11$
9. $x_3^2 + x_4^2 \geq 14$
10. $x_1^2 + x_3^2 + x_4^2 \geq 17$
11. $x_1^2 + x_2^2 + x_3^2 \geq 17$
12. $2x_1 + 10x_3 + 9x_4 \geq 12$
13. $x_1^2 + x_2^2 + x_3^2 \geq 12$
14. $x_1 + 10x_3 \leq 58$
15. $x_1^2 + x_2^2 \leq 49$
16. $x_2 + 10x_3 \leq 27$
17. $10x_3^2 + 9x_4^2 \leq 82$
18. $x_1 + 9x_4 \leq 36$
19. $x_2 + 9x_4 \leq 69$
20. $2x_1 + x_2 + 10x_3 + 9x_4 \leq 69$

## Step 4: Define the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'vincas'), ('x2', 'tomato vines'), ('x3', 'strawberry bushes'), ('x4', 'ferns')],
'objective_function': '4*x1^2 + 9*x1*x2 + 9*x1*x3 + 4*x1*x4 + 9*x2*x3 + 6*x3^2 + 6*x3*x4 + 4*x4^2 + 2*x2',
'constraints': [
'2*x1 <= 92',
'x2 <= 92',
'10*x3 <= 92',
'9*x4 <= 92',
'2*x1 + 9*x4 >= 12',
'x1^2 + x3^2 >= 22',
'x2 + 10*x3 >= 9',
'2*x1 + x2 >= 11',
'x3^2 + x4^2 >= 14',
'x1^2 + x3^2 + x4^2 >= 17',
'x1^2 + x2^2 + x3^2 >= 17',
'2*x1 + 10*x3 + 9*x4 >= 12',
'x1^2 + x2^2 + x3^2 >= 12',
'x1 + 10*x3 <= 58',
'x1^2 + x2^2 <= 49',
'x2 + 10*x3 <= 27',
'10*x3^2 + 9*x4^2 <= 82',
'x1 + 9*x4 <= 36',
'x2 + 9*x4 <= 69',
'2*x1 + x2 + 10*x3 + 9*x4 <= 69'
]
}
```

## Step 5: Write the Gurobi code for the optimization problem
```python
import gurobi

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

# Define the variables
x1 = m.addVar(name="vincas", vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name="tomato_vines", vtype=gurobi.GRB.INTEGER)
x3 = m.addVar(name="strawberry_bushes", vtype=gurobi.GRB.INTEGER)
x4 = m.addVar(name="ferns", vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(4*x1**2 + 9*x1*x2 + 9*x1*x3 + 4*x1*x4 + 9*x2*x3 + 6*x3**2 + 6*x3*x4 + 4*x4**2 + 2*x2, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(2*x1 <= 92)
m.addConstr(x2 <= 92)
m.addConstr(10*x3 <= 92)
m.addConstr(9*x4 <= 92)
m.addConstr(2*x1 + 9*x4 >= 12)
m.addConstr(x1**2 + x3**2 >= 22)
m.addConstr(x2 + 10*x3 >= 9)
m.addConstr(2*x1 + x2 >= 11)
m.addConstr(x3**2 + x4**2 >= 14)
m.addConstr(x1**2 + x3**2 + x4**2 >= 17)
m.addConstr(x1**2 + x2**2 + x3**2 >= 17)
m.addConstr(2*x1 + 10*x3 + 9*x4 >= 12)
m.addConstr(x1**2 + x2**2 + x3**2 >= 12)
m.addConstr(x1 + 10*x3 <= 58)
m.addConstr(x1**2 + x2**2 <= 49)
m.addConstr(x2 + 10*x3 <= 27)
m.addConstr(10*x3**2 + 9*x4**2 <= 82)
m.addConstr(x1 + 9*x4 <= 36)
m.addConstr(x2 + 9*x4 <= 69)
m.addConstr(2*x1 + x2 + 10*x3 + 9*x4 <= 69)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Vincas: ", x1.varValue)
    print("Tomato Vines: ", x2.varValue)
    print("Strawberry Bushes: ", x3.varValue)
    print("Ferns: ", x4.varValue)
else:
    print("The model is infeasible")
```