## Step 1: Define the symbolic representation of the variables
The variables are 'postage stamps', 'staplers', and 'hole punches', which we will represent as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $3.31x_0^2 + 6.11x_0x_1 + 5.37x_1^2 + 7.52x_1x_2 + 2.38x_2^2 + 9.71x_1 + 9.72x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $5x_0 + 13x_1 \geq 78$
- $2x_0 + 26x_2 \geq 143$
- $2x_0 + 20x_1 \geq 137$
- $20x_1^2 + 26x_2^2 \geq 63$
- $22x_0 + 18x_1 \geq 82$
- $22x_0 + 28x_2 \geq 76$
- $22x_0^2 + 18x_1^2 + 28x_2^2 \geq 96$
- $-x_0 + 6x_2 \geq 0$
- $5x_0 + x_2 \leq 238$
- $5x_0 + 13x_1 \leq 258$
- $13x_1 + x_2 \leq 169$
- $5x_0 + 13x_1 + x_2 \leq 169$
- $2x_0 + 20x_1 \leq 418$
- $2x_0 + 20x_1 + 26x_2 \leq 418$
- $22x_0 + 28x_2 \leq 252$
- $22x_0 + 18x_1 + 28x_2 \leq 252$
- $28x_0 + 12x_1 \leq 193$
- $28x_0 + 12x_1 \leq 336$
- $28x_0^2 + 12x_1^2 + 6x_2^2 \leq 311$
- $28x_0 + 12x_1 + 6x_2 \leq 311$

## 4: Write the Gurobi code
```python
import gurobi

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

# Define the variables
x0 = model.addVar(name="postage_stamps", vtype=gurobi.GRB.INTEGER)
x1 = model.addVar(name="staplers", vtype=gurobi.GRB.INTEGER)
x2 = model.addVar(name="hole_punches", vtype=gurobi.GRB.INTEGER)

# Define the objective function
model.setObjective(3.31*x0**2 + 6.11*x0*x1 + 5.37*x1**2 + 7.52*x1*x2 + 2.38*x2**2 + 9.71*x1 + 9.72*x2, gurobi.GRB.MAXIMIZE)

# Add constraints
model.addConstr(5*x0 + 13*x1 >= 78)
model.addConstr(2*x0 + 26*x2 >= 143)
model.addConstr(2*x0 + 20*x1 >= 137)
model.addConstr(20*x1**2 + 26*x2**2 >= 63)
model.addConstr(22*x0 + 18*x1 >= 82)
model.addConstr(22*x0 + 28*x2 >= 76)
model.addConstr(22*x0**2 + 18*x1**2 + 28*x2**2 >= 96)
model.addConstr(-x0 + 6*x2 >= 0)
model.addConstr(5*x0 + x2 <= 238)
model.addConstr(5*x0 + 13*x1 <= 258)
model.addConstr(13*x1 + x2 <= 169)
model.addConstr(5*x0 + 13*x1 + x2 <= 169)
model.addConstr(2*x0 + 20*x1 <= 418)
model.addConstr(2*x0 + 20*x1 + 26*x2 <= 418)
model.addConstr(22*x0 + 28*x2 <= 252)
model.addConstr(22*x0 + 18*x1 + 28*x2 <= 252)
model.addConstr(28*x0 + 12*x1 <= 193)
model.addConstr(28*x0 + 12*x1 <= 336)
model.addConstr(28*x0**2 + 12*x1**2 + 6*x2**2 <= 311)
model.addConstr(28*x0 + 12*x1 + 6*x2 <= 311)

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objval)
    print("Postage stamps: ", x0.varValue)
    print("Staplers: ", x1.varValue)
    print("Hole punches: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 5: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'postage stamps'), ('x1', 'staplers'), ('x2', 'hole punches')],
    'objective_function': '3.31*x0^2 + 6.11*x0*x1 + 5.37*x1^2 + 7.52*x1*x2 + 2.38*x2^2 + 9.71*x1 + 9.72*x2',
    'constraints': [
        '5*x0 + 13*x1 >= 78',
        '2*x0 + 26*x2 >= 143',
        '2*x0 + 20*x1 >= 137',
        '20*x1^2 + 26*x2^2 >= 63',
        '22*x0 + 18*x1 >= 82',
        '22*x0 + 28*x2 >= 76',
        '22*x0^2 + 18*x1^2 + 28*x2^2 >= 96',
        '-x0 + 6*x2 >= 0',
        '5*x0 + x2 <= 238',
        '5*x0 + 13*x1 <= 258',
        '13*x1 + x2 <= 169',
        '5*x0 + 13*x1 + x2 <= 169',
        '2*x0 + 20*x1 <= 418',
        '2*x0 + 20*x1 + 26*x2 <= 418',
        '22*x0 + 28*x2 <= 252',
        '22*x0 + 18*x1 + 28*x2 <= 252',
        '28*x0 + 12*x1 <= 193',
        '28*x0 + 12*x1 <= 336',
        '28*x0^2 + 12*x1^2 + 6*x2^2 <= 311',
        '28*x0 + 12*x1 + 6*x2 <= 311'
    ]
}
```