## Step 1: Define the symbolic representation of the variables
The variables are 'cucumber vines', 'coleus', 'tulips', 'tomato vines' which can be represented symbolically as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $5x_0x_1 + 3x_0x_3 + 5x_1x_3 + x_2^2 + 4x_2x_3 + 2x_3^2 + 7x_0 + x_1 + 6x_2$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $9.65x_0 \leq 283$ (beauty rating of cucumber vines, but it seems this should be an equality or another type of constraint, however, based on the problem it seems we consider individual contributions)
- $5.29x_0 \leq 281$ (yield of cucumber vines)
- $15.49x_1 \leq 283$ (beauty rating of coleus)
- $16.96x_1 \leq 281$ (yield of coleus)
- $5.01x_2 \leq 283$ (beauty rating of tulips)
- $11.65x_2 \leq 281$ (yield of tulips)
- $11.51x_3 \leq 283$ (beauty rating of tomato vines)
- $12.34x_3 \leq 281$ (yield of tomato vines)
- $15.49x_1 + 5.01x_2 \geq 57$ (total beauty rating from coleus and tulips)
- $9.65x_0 + 11.51x_3 \geq 59$ (total beauty rating from cucumber vines and tomato vines)
- $(5.01x_2)^2 + (11.51x_3)^2 \geq 24$ (total combined beauty rating from tulips squared and tomato vines squared)
- $(9.65x_0)^2 + (15.49x_1)^2 + (11.51x_3)^2 \geq 66$ (total combined beauty rating from cucumber vines squared, coleus squared, and tomato vines squared)
- $(9.65x_0)^2 + (5.01x_2)^2 + (11.51x_3)^2 \geq 66$ (total combined beauty rating from cucumber vines squared, tulips squared, and tomato vines squared)
- $9.65x_0 + 15.49x_1 + 11.51x_3 \geq 65$ (total beauty rating from cucumber vines, coleus, and tomato vines)
- $9.65x_0 + 5.01x_2 + 11.51x_3 \geq 65$ (total beauty rating from cucumber vines, tulips, and tomato vines)
- $16.96x_1 + 12.34x_3 \geq 32$ (total yield from coleus and tomato vines)
- $(16.96x_1)^2 + (11.65x_2)^2 \geq 67$ (total yield from coleus squared and tulips squared)
- $11.65x_2 + 12.34x_3 \geq 33$ (total yield from tulips and tomato vines)
- $5.29x_0 + 12.34x_3 \geq 67$ (total yield from cucumber vines and tomato vines)
- $-10x_0 + 2x_1 \geq 0$ (relationship between cucumber vines and coleus)
- $(9.65x_0)^2 + (5.01x_2)^2 \leq 184$ (total combined beauty rating from cucumber vines squared and tulips squared)
- $15.49x_1 + 5.01x_2 \leq 217$ (total combined beauty rating from coleus and tulips)
- $(9.65x_0)^2 + (11.51x_3)^2 \leq 154$ (total combined beauty rating from cucumber vines squared and tomato vines squared)
- $9.65x_0 + 15.49x_1 + 5.01x_2 \leq 160$ (total combined beauty rating from cucumber vines, coleus, and tulips)
- $9.65x_0 + 15.49x_1 + 5.01x_2 + 11.51x_3 \leq 160$ (total combined beauty rating from all)
- $(16.96x_1)^2 + (11.65x_2)^2 \leq 175$ (total yield from coleus squared and tulips squared)
- $5.29x_0 + 16.96x_1 + 11.65x_2 + 12.34x_3 \leq 175$ (total yield from all)

## 4: Convert the problem into Gurobi code
```python
import gurobi

# Define the model
model = gurobi.Model()

# Define the variables
x0 = model.addVar(name="cucumber_vines", vtype=gurobi.GRB.INTEGER)
x1 = model.addVar(name="coleus", vtype=gurobi.GRB.INTEGER)
x2 = model.addVar(name="tulips", vtype=gurobi.GRB.INTEGER)
x3 = model.addVar(name="tomato_vines", vtype=gurobi.GRB.INTEGER)

# Define the objective function
model.setObjective(5*x0*x1 + 3*x0*x3 + 5*x1*x3 + x2**2 + 4*x2*x3 + 2*x3**2 + 7*x0 + x1 + 6*x2, gurobi.GRB.MAXIMIZE)

# Define the constraints
model.addConstr(9.65 * x0 <= 283)
model.addConstr(5.29 * x0 <= 281)
model.addConstr(15.49 * x1 <= 283)
model.addConstr(16.96 * x1 <= 281)
model.addConstr(5.01 * x2 <= 283)
model.addConstr(11.65 * x2 <= 281)
model.addConstr(11.51 * x3 <= 283)
model.addConstr(12.34 * x3 <= 281)

model.addConstr(15.49 * x1 + 5.01 * x2 >= 57)
model.addConstr(9.65 * x0 + 11.51 * x3 >= 59)
model.addConstr((5.01 * x2)**2 + (11.51 * x3)**2 >= 24)
model.addConstr((9.65 * x0)**2 + (15.49 * x1)**2 + (11.51 * x3)**2 >= 66)
model.addConstr((9.65 * x0)**2 + (5.01 * x2)**2 + (11.51 * x3)**2 >= 66)
model.addConstr(9.65 * x0 + 15.49 * x1 + 11.51 * x3 >= 65)
model.addConstr(9.65 * x0 + 5.01 * x2 + 11.51 * x3 >= 65)
model.addConstr(16.96 * x1 + 12.34 * x3 >= 32)
model.addConstr((16.96 * x1)**2 + (11.65 * x2)**2 >= 67)
model.addConstr(11.65 * x2 + 12.34 * x3 >= 33)
model.addConstr(5.29 * x0 + 12.34 * x3 >= 67)
model.addConstr(-10 * x0 + 2 * x1 >= 0)
model.addConstr((9.65 * x0)**2 + (5.01 * x2)**2 <= 184)
model.addConstr(15.49 * x1 + 5.01 * x2 <= 217)
model.addConstr((9.65 * x0)**2 + (11.51 * x3)**2 <= 154)
model.addConstr(9.65 * x0 + 15.49 * x1 + 5.01 * x2 <= 160)
model.addConstr(9.65 * x0 + 15.49 * x1 + 5.01 * x2 + 11.51 * x3 <= 160)
model.addConstr((16.96 * x1)**2 + (11.65 * x2)**2 <= 175)
model.addConstr(5.29 * x0 + 16.96 * x1 + 11.65 * x2 + 12.34 * x3 <= 175)

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objVal)
    print("Cucumber Vines: ", x0.varValue)
    print("Coleus: ", x1.varValue)
    print("Tulips: ", x2.varValue)
    print("Tomato Vines: ", x3.varValue)
else:
    print("The model is infeasible")
```

## Step 5: Provide the symbolic representation
```json
{
    'sym_variables': [('x0', 'cucumber vines'), ('x1', 'coleus'), ('x2', 'tulips'), ('x3', 'tomato vines')],
    'objective_function': '5*x0*x1 + 3*x0*x3 + 5*x1*x3 + x2^2 + 4*x2*x3 + 2*x3^2 + 7*x0 + x1 + 6*x2',
    'constraints': [
        '9.65*x0 <= 283',
        '5.29*x0 <= 281',
        '15.49*x1 <= 283',
        '16.96*x1 <= 281',
        '5.01*x2 <= 283',
        '11.65*x2 <= 281',
        '11.51*x3 <= 283',
        '12.34*x3 <= 281',
        '15.49*x1 + 5.01*x2 >= 57',
        '9.65*x0 + 11.51*x3 >= 59',
        '(5.01*x2)^2 + (11.51*x3)^2 >= 24',
        '(9.65*x0)^2 + (15.49*x1)^2 + (11.51*x3)^2 >= 66',
        '(9.65*x0)^2 + (5.01*x2)^2 + (11.51*x3)^2 >= 66',
        '9.65*x0 + 15.49*x1 + 11.51*x3 >= 65',
        '9.65*x0 + 5.01*x2 + 11.51*x3 >= 65',
        '16.96*x1 + 12.34*x3 >= 32',
        '(16.96*x1)^2 + (11.65*x2)^2 >= 67',
        '11.65*x2 + 12.34*x3 >= 33',
        '5.29*x0 + 12.34*x3 >= 67',
        '-10*x0 + 2*x1 >= 0',
        '(9.65*x0)^2 + (5.01*x2)^2 <= 184',
        '15.49*x1 + 5.01*x2 <= 217',
        '(9.65*x0)^2 + (11.51*x3)^2 <= 154',
        '9.65*x0 + 15.49*x1 + 5.01*x2 <= 160',
        '9.65*x0 + 15.49*x1 + 5.01*x2 + 11.51*x3 <= 160',
        '(16.96*x1)^2 + (11.65*x2)^2 <= 175',
        '5.29*x0 + 16.96*x1 + 11.65*x2 + 12.34*x3 <= 175'
    ]
}
```