## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'corn cobs' and 'black beans', which we can denote as $x_1$ and $x_2$ respectively. The objective function to maximize is $9x_1^2 + 9x_1x_2 + 4x_2^2 + 9x_1 + 8x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
- $7x_1 + 6x_2 \leq 41$ (cost constraint)
- $x_1^2 + x_2^2 \geq 6$ (cost squared constraint)
- $x_1 + 4x_2 \geq 12$ (umami index constraint)
- $5x_1 + 7x_2 \geq 20$ (fat constraint)
- $7x_1 + 2x_2 \geq 39$ (fiber constraint)
- $4x_1^2 + 4x_2^2 \geq 17$ (sourness index squared constraint)
- $-4x_1 + x_2 \geq 0$ (linear constraint)
- $7x_1 + 6x_2 \leq 22$ (alternate cost constraint, seems redundant with the first cost constraint)
- $x_1 + 4x_2 \leq 16$ (umami index upper bound)
- $5x_1 + 7x_2 \leq 54$ (fat upper bound)
- $7x_1 + 2x_2 \leq 71$ (fiber upper bound, seems to incorrectly suggest squared terms)
- $4x_1^2 + 4x_2^2 \leq 27$ (sourness index squared upper bound)
- $4x_1 + 4x_2 \leq 27$ (sourness index upper bound)
- $x_1$ is an integer (integrality constraint)
- $x_2$ can be a non-integer

## 3: Correct and Clarify Constraints
Upon review, there seems to be some redundancy and potential for error in the constraints, particularly regarding the cost and the representation of constraints involving squared terms. The problem statement suggests:
- Cost: $7x_1 + 6x_2 \leq 41$
- Other constraints as listed, with careful attention to which are actually squared and which are not.

## 4: Formulate the Symbolic Representation
The symbolic representation includes:
- Variables: $x_1$ for 'corn cobs', $x_2$ for 'black beans'
- Objective Function: $9x_1^2 + 9x_1x_2 + 4x_2^2 + 9x_1 + 8x_2$
- Constraints:
  1. $7x_1 + 6x_2 \leq 41$
  2. $x_1^2 + x_2^2 \geq 6$
  3. $x_1 + 4x_2 \geq 12$
  4. $5x_1 + 7x_2 \geq 20$
  5. $7x_1 + 2x_2 \geq 39$
  6. $4(x_1^2 + x_2^2) \geq 17$
  7. $-4x_1 + x_2 \geq 0$
  8. $x_1 + 4x_2 \leq 16$
  9. $5x_1 + 7x_2 \leq 54$
  10. $7x_1 + 2x_2 \leq 71$
  11. $4(x_1^2 + x_2^2) \leq 27$
  12. $4x_1 + 4x_2 \leq 27$

## 5: Gurobi Code Implementation
```python
import gurobi

# Define the model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name='corn_cobs', vtype='I')  # Integer variable for corn cobs
x2 = m.addVar(name='black_beans')  # Continuous variable for black beans

# Objective function
m.setObjective(9*x1**2 + 9*x1*x2 + 4*x2**2 + 9*x1 + 8*x2, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(7*x1 + 6*x2 <= 41)  # Cost constraint
m.addConstr(x1**2 + x2**2 >= 6)  # Cost squared constraint
m.addConstr(x1 + 4*x2 >= 12)  # Umami index constraint
m.addConstr(5*x1 + 7*x2 >= 20)  # Fat constraint
m.addConstr(7*x1 + 2*x2 >= 39)  # Fiber constraint
m.addConstr(4*(x1**2 + x2**2) >= 17)  # Sourness index squared constraint
m.addConstr(-4*x1 + x2 >= 0)  # Linear constraint
m.addConstr(x1 + 4*x2 <= 16)  # Umami index upper bound
m.addConstr(5*x1 + 7*x2 <= 54)  # Fat upper bound
m.addConstr(7*x1 + 2*x2 <= 71)  # Fiber upper bound
m.addConstr(4*(x1**2 + x2**2) <= 27)  # Sourness index squared upper bound
m.addConstr(4*x1 + 4*x2 <= 27)  # Sourness index upper bound

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objVal)
    print('Corn Cobs: ', x1.varValue)
    print('Black Beans: ', x2.varValue)
else:
    print('No solution found')
```

## 6: Symbolic Representation Output
```json
{
    'sym_variables': [('x1', 'corn cobs'), ('x2', 'black beans')],
    'objective_function': '9*x1^2 + 9*x1*x2 + 4*x2^2 + 9*x1 + 8*x2',
    'constraints': [
        '7*x1 + 6*x2 <= 41',
        'x1^2 + x2^2 >= 6',
        'x1 + 4*x2 >= 12',
        '5*x1 + 7*x2 >= 20',
        '7*x1 + 2*x2 >= 39',
        '4*(x1^2 + x2^2) >= 17',
        '-4*x1 + x2 >= 0',
        'x1 + 4*x2 <= 16',
        '5*x1 + 7*x2 <= 54',
        '7*x1 + 2*x2 <= 71',
        '4*(x1^2 + x2^2) <= 27',
        '4*x1 + 4*x2 <= 27'
    ]
}
```