## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- $x_1$ represents 'lemons'
- $x_2$ represents 'potatoes'
- $x_3$ represents 'ravioli'
- $x_4$ represents 'black beans'

## Step 2: Translate the objective function into symbolic notation
The objective function to maximize is: $7.44x_1^2 + 5.02x_1x_2 + 1.88x_1x_3 + 2.78x_2 + 7.87x_3 + 8.79x_4$

## 3: Define the constraints in symbolic notation
The constraints are:
- $11x_1 + 20x_2 + 15x_3 + 4x_4 \leq 246$ (total carbohydrates constraint)
- $20x_2^2 + 15x_3^2 \geq 29$ (carbohydrates from potatoes and ravioli squared)
- $11x_1 + 15x_3 \geq 37$ (carbohydrates from lemons and ravioli)
- $15x_3 + 4x_4 \leq 86$ (carbohydrates from ravioli and black beans)
- $20x_2 + 4x_4 \leq 218$ (carbohydrates from potatoes and black beans)
- $11^2x_1^2 + 20^2x_2^2 \leq 185$ (carbohydrates from lemons squared and potatoes squared)
- $11x_1 + 20x_2 + 15x_3 + 4x_4 \leq 185$ (total carbohydrates from all sources)

## 4: Create the Gurobi model and variables
We will create a Gurobi model and define the variables $x_1, x_2, x_3, x_4$ with no bounds specified as the problem allows for fractional amounts.

## 5: Implement the objective function and constraints in Gurobi
```python
import gurobi

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

# Define the variables
x1 = model.addVar(name="lemons", lb=0) # Assuming no negative amounts
x2 = model.addVar(name="potatoes", lb=0)
x3 = model.addVar(name="ravioli", lb=0)
x4 = model.addVar(name="black_beans", lb=0)

# Objective function
model.setObjective(7.44 * x1**2 + 5.02 * x1 * x2 + 1.88 * x1 * x3 + 2.78 * x2 + 7.87 * x3 + 8.79 * x4, gurobi.GRB.MAXIMIZE)

# Constraints
model.addConstr(11 * x1 + 20 * x2 + 15 * x3 + 4 * x4 <= 246, name="carbohydrates")
model.addConstr(20 * x2**2 + 15 * x3**2 >= 29, name="potatoes_ravioli_squared")
model.addConstr(11 * x1 + 15 * x3 >= 37, name="lemons_ravioli")
model.addConstr(15 * x3 + 4 * x4 <= 86, name="ravioli_black_beans")
model.addConstr(20 * x2 + 4 * x4 <= 218, name="potatoes_black_beans")
model.addConstr(121 * x1**2 + 400 * x2**2 <= 185, name="lemons_squared_potatoes_squared")
model.addConstr(11 * x1 + 20 * x2 + 15 * x3 + 4 * x4 <= 185, name="total_carbohydrates")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objVal)
    print("Lemons: ", x1.varValue)
    print("Potatoes: ", x2.varValue)
    print("Ravioli: ", x3.varValue)
    print("Black Beans: ", x4.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'lemons'), ('x2', 'potatoes'), ('x3', 'ravioli'), ('x4', 'black beans')],
    'objective_function': '7.44x1^2 + 5.02x1x2 + 1.88x1x3 + 2.78x2 + 7.87x3 + 8.79x4',
    'constraints': [
        '11x1 + 20x2 + 15x3 + 4x4 <= 246',
        '20x2^2 + 15x3^2 >= 29',
        '11x1 + 15x3 >= 37',
        '15x3 + 4x4 <= 86',
        '20x2 + 4x4 <= 218',
        '121x1^2 + 400x2^2 <= 185',
        '11x1 + 20x2 + 15x3 + 4x4 <= 185'
    ]
}
```

```python
import gurobi

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

# Define the variables
x1 = model.addVar(name="lemons", lb=0) 
x2 = model.addVar(name="potatoes", lb=0)
x3 = model.addVar(name="ravioli", lb=0)
x4 = model.addVar(name="black_beans", lb=0)

# Objective function
model.setObjective(7.44 * x1**2 + 5.02 * x1 * x2 + 1.88 * x1 * x3 + 2.78 * x2 + 7.87 * x3 + 8.79 * x4, gurobi.GRB.MAXIMIZE)

# Constraints
model.addConstr(11 * x1 + 20 * x2 + 15 * x3 + 4 * x4 <= 246, name="carbohydrates")
model.addConstr(20 * x2**2 + 15 * x3**2 >= 29, name="potatoes_ravioli_squared")
model.addConstr(11 * x1 + 15 * x3 >= 37, name="lemons_ravioli")
model.addConstr(15 * x3 + 4 * x4 <= 86, name="ravioli_black_beans")
model.addConstr(20 * x2 + 4 * x4 <= 218, name="potatoes_black_beans")
model.addConstr(121 * x1**2 + 400 * x2**2 <= 185, name="lemons_squared_potatoes_squared")
model.addConstr(11 * x1 + 20 * x2 + 15 * x3 + 4 * x4 <= 185, name="total_carbohydrates")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objVal)
    print("Lemons: ", x1.varValue)
    print("Potatoes: ", x2.varValue)
    print("Ravioli: ", x3.varValue)
    print("Black Beans: ", x4.varValue)
else:
    print("The model is infeasible")
```