## Step 1: Define the symbolic representation of the variables
The variables are ['milligrams of vitamin A', 'grams of fat', 'grams of fiber']. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables is:
- $x_1$ = 'milligrams of vitamin A'
- $x_2$ = 'grams of fat'
- $x_3$ = 'grams of fiber'

## 3: Define the objective function in symbolic notation
The objective function to maximize is $1.97x_1^2 + 1.23x_3^2 + 5.84x_3$.

## 4: List the constraints in symbolic notation
The constraints are:
- $14x_1 \leq 218$
- $20x_1 \leq 88$
- $17x_2 \leq 218$
- $8x_2 \leq 88$
- $15x_3 \leq 218$
- $19x_3 \leq 88$
- $14x_1 + 17x_2 \geq 54$
- $14x_1 + 15x_3 \geq 26$
- $14x_1 + 17x_2 + 15x_3 \geq 44$
- $14x_1 + 15x_3 \leq 178$
- $17x_2 + 15x_3 \leq 132$
- $14x_1 + 17x_2 + 15x_3 \leq 152$
- $14x_1 + 8x_2 \leq 78$
- $8x_2^2 + 19x_3^2 \leq 56$
- $20x_1 + 8x_2 + 19x_3 \leq 56$
- $x_1$ is an integer
- $x_2$ is an integer

## 5: Create the Gurobi model and variables
We will create a Gurobi model and define the variables $x_1$, $x_2$, and $x_3$.

## 6: Implement the objective function and constraints in Gurobi
We will use Gurobi's Python API to implement the objective function and constraints.

## 7: Solve the optimization problem
We will solve the optimization problem using Gurobi's solver.

## 8: Output the solution
We will output the symbolic representation of the problem and the Gurobi code.

```json
{
    'sym_variables': [
        ('x1', 'milligrams of vitamin A'), 
        ('x2', 'grams of fat'), 
        ('x3', 'grams of fiber')
    ], 
    'objective_function': '1.97*x1^2 + 1.23*x3^2 + 5.84*x3', 
    'constraints': [
        '14*x1 <= 218', 
        '20*x1 <= 88', 
        '17*x2 <= 218', 
        '8*x2 <= 88', 
        '15*x3 <= 218', 
        '19*x3 <= 88', 
        '14*x1 + 17*x2 >= 54', 
        '14*x1 + 15*x3 >= 26', 
        '14*x1 + 17*x2 + 15*x3 >= 44', 
        '14*x1 + 15*x3 <= 178', 
        '17*x2 + 15*x3 <= 132', 
        '14*x1 + 17*x2 + 15*x3 <= 152', 
        '14*x1 + 8*x2 <= 78', 
        '8*x2^2 + 19*x3^2 <= 56', 
        '20*x1 + 8*x2 + 19*x3 <= 56'
    ]
}
```

```python
import gurobi as gp

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

# Define the variables
x1 = m.addVar(name="milligrams of vitamin A", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="grams of fat", vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="grams of fiber")

# Set the objective function
m.setObjective(1.97 * x1 ** 2 + 1.23 * x3 ** 2 + 5.84 * x3, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(14 * x1 <= 218)
m.addConstr(20 * x1 <= 88)
m.addConstr(17 * x2 <= 218)
m.addConstr(8 * x2 <= 88)
m.addConstr(15 * x3 <= 218)
m.addConstr(19 * x3 <= 88)
m.addConstr(14 * x1 + 17 * x2 >= 54)
m.addConstr(14 * x1 + 15 * x3 >= 26)
m.addConstr(14 * x1 + 17 * x2 + 15 * x3 >= 44)
m.addConstr(14 * x1 + 15 * x3 <= 178)
m.addConstr(17 * x2 + 15 * x3 <= 132)
m.addConstr(14 * x1 + 17 * x2 + 15 * x3 <= 152)
m.addConstr(14 * x1 + 8 * x2 <= 78)
m.addConstr(8 * x2 ** 2 + 19 * x3 ** 2 <= 56)
m.addConstr(20 * x1 + 8 * x2 + 19 * x3 <= 56)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print("Milligrams of vitamin A: ", x1.varValue)
    print("Grams of fat: ", x2.varValue)
    print("Grams of fiber: ", x3.varValue)
    print("Objective function value: ", m.objVal)
else:
    print("No optimal solution found.")
```