## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

## Step 2: Identify the variables and their corresponding symbolic representations
The variables are 'fruit salads', 'strawberries', and 'peanutbutter sandwiches', which we can represent symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## 3: Formulate the objective function
The objective function to maximize is $1x_0^2 + 3x_0x_1 + 5x_0x_2 + 9x_1^2 + 8x_0 + 8x_1$.

## 4: List the constraints
The constraints given are:
- $6x_0 \geq 68$ is not correct, it seems there was a misunderstanding. The correct interpretation is that the umami index of fruit salads is 6, which is a fixed value, not a constraint in the traditional sense. However, we have:
- $6x_0 + 15x_1 \geq 68$
- $15x_1 + 5x_2 \geq 37$
- $6x_0 + 5x_2 \geq 27$
- $5x_0 + 19x_2 \geq 78$
- $5x_0 + x_1 \geq 40$
- $15x_0^2 + 7x_1^2 \geq 49$
- $7x_1 + 12x_2 \geq 88$
- $15x_0 + 12x_2 \geq 138$
- $15x_1 + 5x_2 \leq 196$
- $6x_0^2 + 5x_2^2 \leq 131$
- $6x_0 + 15x_1 + 5x_2 \leq 160$
- $8x_0 + 20x_2 \leq 140$
- $8x_0 + 5x_1 \leq 122$
- $8x_0 + 5x_1 + 20x_2 \leq 122$
- $5x_0^2 + x_1^2 \leq 232$
- $5x_0 + x_1 + 19x_2 \leq 232$
- $7x_1^2 + 12x_2^2 \leq 266$
- $15x_0^2 + 7x_1^2 \leq 160$
- $15x_0 + 7x_1 + 12x_2 \leq 160$
- $x_0, x_1, x_2$ are integers.

## 5: Convert the problem into a Gurobi code
Now, let's write the Gurobi code for this problem.

```python
import gurobi

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

# Define the variables
x0 = m.addVar(name="fruit_salads", vtype=gurobi.GRB.INTEGER)
x1 = m.addVar(name="strawberries", vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name="peanutbutter_sandwiches", vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(1*x0**2 + 3*x0*x1 + 5*x0*x2 + 9*x1**2 + 8*x0 + 8*x1, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(6*x0 + 15*x1 >= 68)
m.addConstr(15*x1 + 5*x2 >= 37)
m.addConstr(6*x0 + 5*x2 >= 27)
m.addConstr(5*x0 + 19*x2 >= 78)
m.addConstr(5*x0 + x1 >= 40)
m.addConstr(15*x0**2 + 7*x1**2 >= 49)
m.addConstr(7*x1 + 12*x2 >= 88)
m.addConstr(15*x0 + 12*x2 >= 138)
m.addConstr(15*x1 + 5*x2 <= 196)
m.addConstr(6*x0**2 + 5*x2**2 <= 131)
m.addConstr(6*x0 + 15*x1 + 5*x2 <= 160)
m.addConstr(8*x0 + 20*x2 <= 140)
m.addConstr(8*x0 + 5*x1 <= 122)
m.addConstr(8*x0 + 5*x1 + 20*x2 <= 122)
m.addConstr(5*x0**2 + x1**2 <= 232)
m.addConstr(5*x0 + x1 + 19*x2 <= 232)
m.addConstr(7*x1**2 + 12*x2**2 <= 266)
m.addConstr(15*x0**2 + 7*x1**2 <= 160)
m.addConstr(15*x0 + 7*x1 + 12*x2 <= 160)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Fruit Salads: ", x0.varValue)
    print("Strawberries: ", x1.varValue)
    print("Peanutbutter Sandwiches: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic Representation
The symbolic representation of the problem is as follows:

```json
{
    'sym_variables': [
        ('x0', 'fruit salads'),
        ('x1', 'strawberries'),
        ('x2', 'peanutbutter sandwiches')
    ],
    'objective_function': '1*x0^2 + 3*x0*x1 + 5*x0*x2 + 9*x1^2 + 8*x0 + 8*x1',
    'constraints': [
        '6*x0 + 15*x1 >= 68',
        '15*x1 + 5*x2 >= 37',
        '6*x0 + 5*x2 >= 27',
        '5*x0 + 19*x2 >= 78',
        '5*x0 + x1 >= 40',
        '15*x0^2 + 7*x1^2 >= 49',
        '7*x1 + 12*x2 >= 88',
        '15*x0 + 12*x2 >= 138',
        '15*x1 + 5*x2 <= 196',
        '6*x0^2 + 5*x2^2 <= 131',
        '6*x0 + 15*x1 + 5*x2 <= 160',
        '8*x0 + 20*x2 <= 140',
        '8*x0 + 5*x1 <= 122',
        '8*x0 + 5*x1 + 20*x2 <= 122',
        '5*x0^2 + x1^2 <= 232',
        '5*x0 + x1 + 19*x2 <= 232',
        '7*x1^2 + 12*x2^2 <= 266',
        '15*x0^2 + 7*x1^2 <= 160',
        '15*x0 + 7*x1 + 12*x2 <= 160'
    ]
}
```