## Step 1: Define the variables and their attributes
We have four variables: 'chives', 'agave', 'squash plants', and 'carrots'. Each variable has attributes for water need, growth speed, and beauty rating.

## Step 2: Formulate the objective function
The objective function to maximize is: $6 \cdot chives^2 + 7 \cdot chives \cdot carrots + 4 \cdot agave \cdot squash\ plants + 7 \cdot squash\ plants + 2 \cdot carrots$.

## 3: Define the constraints
1. $14 \cdot chives + 8 \cdot agave + 7 \cdot squash\ plants + 11 \cdot carrots \leq 154$
2. $12 \cdot chives + 13 \cdot agave + 11 \cdot squash\ plants + 6 \cdot carrots \leq 103$
3. $10 \cdot chives + 4 \cdot agave + 14 \cdot squash\ plants + 6 \cdot carrots \leq 215$
4. $8 \cdot agave + 11 \cdot carrots \geq 28$
5. $12 \cdot chives + 13 \cdot agave + 11 \cdot squash\ plants \geq 16$
6. $12^2 \cdot chives + 13^2 \cdot agave + 6^2 \cdot carrots \geq 16$
7. $12^2 \cdot chives + 13^2 \cdot agave + 11^2 \cdot squash\ plants \geq 25$
8. $10^2 \cdot chives + 4^2 \cdot agave + 6^2 \cdot carrots \geq 47$
9. $14 \cdot squash\ plants + 6 \cdot carrots \geq 22$
10. $4 \cdot agave + 14 \cdot squash\ plants \geq 21$
11. $4^2 \cdot agave + 14^2 \cdot squash\ plants + 6^2 \cdot carrots \geq 33$
12. $-4 \cdot squash\ plants^2 + 1 \cdot carrots^2 \geq 0$
13. $14^2 \cdot chives + 7^2 \cdot squash\ plants \leq 69$
14. $14 \cdot chives + 8 \cdot agave \leq 42$
15. $14 \cdot chives + 11 \cdot carrots \leq 92$
16. $7 \cdot squash\ plants + 11 \cdot carrots \leq 62$
17. $14^2 \cdot chives + 8^2 \cdot agave + 11^2 \cdot carrots \leq 135$
18. $14 \cdot chives + 8 \cdot agave + 7 \cdot squash\ plants + 11 \cdot carrots \leq 135$
19. $12 \cdot chives + 11 \cdot squash\ plants \leq 71$
20. $13 \cdot agave + 6 \cdot carrots \leq 77$
21. $12 \cdot chives + 13 \cdot agave + 11 \cdot squash\ plants \leq 45$
22. $12 \cdot chives + 13 \cdot agave + 11 \cdot squash\ plants + 6 \cdot carrots \leq 45$
23. $10 \cdot chives + 4 \cdot agave \leq 120$
24. $14 \cdot squash\ plants + 6 \cdot carrots \leq 161$
25. $4 \cdot agave + 14 \cdot squash\ plants \leq 56$
26. $10^2 \cdot chives + 4^2 \cdot agave + 14^2 \cdot squash\ plants \leq 160$
27. $10 \cdot chives + 14 \cdot squash\ plants + 6 \cdot carrots \leq 174$
28. $10 \cdot chives + 4 \cdot agave + 14 \cdot squash\ plants + 6 \cdot carrots \leq 174$

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

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

# Define the variables
chives = m.addVar(name="chives", vtype=gurobi.GRB.INTEGER)
agave = m.addVar(name="agave", vtype=gurobi.GRB.INTEGER)
squash_plants = m.addVar(name="squash_plants", vtype=gurobi.GRB.INTEGER)
carrots = m.addVar(name="carrots", vtype=gurobi.GRB.INTEGER)

# Define the objective function
m.setObjective(6 * chives**2 + 7 * chives * carrots + 4 * agave * squash_plants + 7 * squash_plants + 2 * carrots, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(14 * chives + 8 * agave + 7 * squash_plants + 11 * carrots <= 154)
m.addConstr(12 * chives + 13 * agave + 11 * squash_plants + 6 * carrots <= 103)
m.addConstr(10 * chives + 4 * agave + 14 * squash_plants + 6 * carrots <= 215)
m.addConstr(8 * agave + 11 * carrots >= 28)
m.addConstr(12 * chives + 13 * agave + 11 * squash_plants >= 16)
m.addConstr(12**2 * chives + 13**2 * agave + 6**2 * carrots >= 16)
m.addConstr(12**2 * chives + 13**2 * agave + 11**2 * squash_plants >= 25)
m.addConstr(10**2 * chives + 4**2 * agave + 6**2 * carrots >= 47)
m.addConstr(14 * squash_plants + 6 * carrots >= 22)
m.addConstr(4 * agave + 14 * squash_plants >= 21)
m.addConstr(4**2 * agave + 14**2 * squash_plants + 6**2 * carrots >= 33)
m.addConstr(-4 * squash_plants**2 + 1 * carrots**2 >= 0)
m.addConstr(14**2 * chives + 7**2 * squash_plants <= 69)
m.addConstr(14 * chives + 8 * agave <= 42)
m.addConstr(14 * chives + 11 * carrots <= 92)
m.addConstr(7 * squash_plants + 11 * carrots <= 62)
m.addConstr(14**2 * chives + 8**2 * agave + 11**2 * carrots <= 135)
m.addConstr(14 * chives + 8 * agave + 7 * squash_plants + 11 * carrots <= 135)
m.addConstr(12 * chives + 11 * squash_plants <= 71)
m.addConstr(13 * agave + 6 * carrots <= 77)
m.addConstr(12 * chives + 13 * agave + 11 * squash_plants <= 45)
m.addConstr(12 * chives + 13 * agave + 11 * squash_plants + 6 * carrots <= 45)
m.addConstr(10 * chives + 4 * agave <= 120)
m.addConstr(14 * squash_plants + 6 * carrots <= 161)
m.addConstr(4 * agave + 14 * squash_plants <= 56)
m.addConstr(10**2 * chives + 4**2 * agave + 14**2 * squash_plants <= 160)
m.addConstr(10 * chives + 14 * squash_plants + 6 * carrots <= 174)
m.addConstr(10 * chives + 4 * agave + 14 * squash_plants + 6 * carrots <= 174)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Chives: ", chives.varValue)
    print("Agave: ", agave.varValue)
    print("Squash Plants: ", squash_plants.varValue)
    print("Carrots: ", carrots.varValue)
else:
    print("The model is infeasible")
```