## Step 1: Define the symbolic representation of the variables
The variables are 'bananas', 'chicken thighs', 'protein bars', 'rotisserie chickens' which can be represented symbolically as $x_1, x_2, x_3, x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is: $4.62x_1^2 + 1.3x_1x_2 + 8.08x_1x_3 + 7.51x_1x_4 + 3.8x_2^2 + 1.42x_2x_3 + 9.75x_2x_4 + 8.05x_3^2 + 4.96x_3x_4 + 3.83x_4^2 + 7.94x_1 + 1.99x_2 + 3.79x_3$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $x_1 \geq 0$, $x_2 \geq 0$, $x_3 \geq 0$, $x_4 \geq 0$ (non-negativity constraints, implicitly handled by Gurobi)
- $9x_2^2 + 8x_4^2 \geq 43$ (at least 43 grams of fiber from chicken thighs squared plus rotisserie chickens squared)
- $x_1 + 9x_2 + 3x_3 \geq 54$ (at least 54 grams of fiber from bananas, chicken thighs, and protein bars)
- $x_1 + 3x_3 + 8x_4 \geq 54$ (at least 54 grams of fiber from bananas, protein bars, and rotisserie chickens)
- $x_1 + 9x_2 + 3x_3 \geq 46$ (at least 46 grams of fiber from bananas plus chicken thighs plus protein bars)
- $x_1 + 3x_3 + 8x_4 \geq 46$ (at least 46 grams of fiber from bananas, protein bars, and rotisserie chickens)
- $9x_2^2 + 3x_3^2 \leq 130$ (at most 130 grams of fiber from chicken thighs squared plus protein bars squared)
- $x_1^2 + 3x_3^2 \leq 109$ (up to 109 grams of fiber from bananas squared plus protein bars squared)
- $x_1 + 8x_4 \leq 230$ (no more than 230 grams of fiber from bananas plus rotisserie chickens)
- $3x_3 + 8x_4 \leq 203$ (no more than 203 grams of fiber from protein bars plus rotisserie chickens)
- $9x_2 + 8x_4 \leq 78$ (up to 78 grams of fiber from chicken thighs and rotisserie chickens)
- $x_1^2 + 9x_2^2 \leq 180$ (no more than 180 grams of fiber from bananas squared plus chicken thighs squared)
- $x_1 + 9x_2 + 3x_3 + 8x_4 \leq 180$ (up to 180 grams of fiber from bananas, chicken thighs, protein bars, and rotisserie chickens)

## 4: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x1', 'bananas'), ('x2', 'chicken thighs'), ('x3', 'protein bars'), ('x4', 'rotisserie chickens')],
    'objective_function': '4.62x1^2 + 1.3x1x2 + 8.08x1x3 + 7.51x1x4 + 3.8x2^2 + 1.42x2x3 + 9.75x2x4 + 8.05x3^2 + 4.96x3x4 + 3.83x4^2 + 7.94x1 + 1.99x2 + 3.79x3',
    'constraints': [
        '9x2^2 + 8x4^2 >= 43',
        'x1 + 9x2 + 3x3 >= 54',
        'x1 + 3x3 + 8x4 >= 54',
        'x1 + 9x2 + 3x3 >= 46',
        'x1 + 3x3 + 8x4 >= 46',
        '9x2^2 + 3x3^2 <= 130',
        'x1^2 + 3x3^2 <= 109',
        'x1 + 8x4 <= 230',
        '3x3 + 8x4 <= 203',
        '9x2 + 8x4 <= 78',
        'x1^2 + 9x2^2 <= 180',
        'x1 + 9x2 + 3x3 + 8x4 <= 180'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

# Define the model
model = gurobi.Model()

# Define the variables
bananas = model.addVar(name="bananas", lb=0)
chicken_thighs = model.addVar(name="chicken thighs", lb=0)
protein_bars = model.addVar(name="protein bars", lb=0)
rotisserie_chickens = model.addVar(name="rotisserie chickens", lb=0)

# Define the objective function
model.setObjective(4.62 * bananas**2 + 1.3 * bananas * chicken_thighs + 8.08 * bananas * protein_bars + 7.51 * bananas * rotisserie_chickens +
                   3.8 * chicken_thighs**2 + 1.42 * chicken_thighs * protein_bars + 9.75 * chicken_thighs * rotisserie_chickens +
                   8.05 * protein_bars**2 + 4.96 * protein_bars * rotisserie_chickens + 3.83 * rotisserie_chickens**2 +
                   7.94 * bananas + 1.99 * chicken_thighs + 3.79 * protein_bars, gurobi.GRB.MAXIMIZE)

# Define the constraints
model.addConstr(9 * chicken_thighs**2 + 8 * rotisserie_chickens**2 >= 43)
model.addConstr(bananas + 9 * chicken_thighs + 3 * protein_bars >= 54)
model.addConstr(bananas + 3 * protein_bars + 8 * rotisserie_chickens >= 54)
model.addConstr(bananas + 9 * chicken_thighs + 3 * protein_bars >= 46)
model.addConstr(bananas + 3 * protein_bars + 8 * rotisserie_chickens >= 46)
model.addConstr(9 * chicken_thighs**2 + 3 * protein_bars**2 <= 130)
model.addConstr(bananas**2 + 3 * protein_bars**2 <= 109)
model.addConstr(bananas + 8 * rotisserie_chickens <= 230)
model.addConstr(3 * protein_bars + 8 * rotisserie_chickens <= 203)
model.addConstr(9 * chicken_thighs + 8 * rotisserie_chickens <= 78)
model.addConstr(bananas**2 + 9 * chicken_thighs**2 <= 180)
model.addConstr(bananas + 9 * chicken_thighs + 3 * protein_bars + 8 * rotisserie_chickens <= 180)

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Bananas: {bananas.varValue}")
    print(f"Chicken Thighs: {chicken_thighs.varValue}")
    print(f"Protein Bars: {protein_bars.varValue}")
    print(f"Rotisserie Chickens: {rotisserie_chickens.varValue}")
    print(f"Objective: {model.objVal}")
else:
    print("No optimal solution found.")
```