## Step 1: Define the symbolic representation of the variables
The variables are ['milligrams of zinc', 'milligrams of vitamin E', 'milligrams of vitamin B6']. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables is:
- $x_0$ : milligrams of zinc
- $x_1$ : milligrams of vitamin E
- $x_2$ : milligrams of vitamin B6

## 3: Define the objective function in symbolic notation
The objective function to minimize is: $3x_0^2 + 2x_0x_1 + 5x_0x_2 + 4x_1^2 + 4x_2^2 + 3x_0 + 2x_2$

## 4: List the constraints in symbolic notation
The constraints are:
- $15x_0 \leq 320$
- $19x_0 \leq 222$
- $13x_0 \leq 285$
- $15x_1 \leq 320$
- $8x_1 \leq 222$
- $3x_1 \leq 285$
- $20x_2 \leq 320$
- $9x_2 \leq 222$
- $16x_2 \leq 285$
- $15x_0 + 15x_1 \geq 49$
- $15^2x_1 + 20^2x_2 \geq 106$
- $15x_0 + 15x_1 + 20x_2 \geq 106$
- $19x_0 + 8x_1 \geq 66$
- $8^2x_1 + 9^2x_2 \geq 46$
- $19x_0 + 8x_1 + 9x_2 \geq 46$
- $3x_1 + 16x_2 \geq 37$
- $13x_0 + 16x_2 \geq 71$
- $13x_0 + 3x_1 + 16x_2 \geq 71$
- $10x_0^2 - 3x_1^2 \geq 0$
- $8x_0 - 8x_2 \geq 0$
- $15x_0 + 20x_2 \leq 150$
- $15x_1 + 20x_2 \leq 270$
- $15x_0 + 15x_1 \leq 320$
- $8x_1 + 9x_2 \leq 94$
- $19x_0 + 9x_2 \leq 205$
- $13^2x_0 + 16^2x_2 \leq 263$
- $3x_1 + 16x_2 \leq 121$
- $x_0 \in \mathbb{Z}$
- $x_1 \in \mathbb{Z}$

## 5: Create the Gurobi model and variables
We will now create a Gurobi model and define the variables.

## 6: Implement the objective function and constraints in Gurobi
```python
import gurobi

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

# Define the variables
x0 = model.addVar(name='milligrams_of_zinc', vtype='I')  # Integer
x1 = model.addVar(name='milligrams_of_vitamin_E', vtype='I')  # Integer
x2 = model.addVar(name='milligrams_of_vitamin_B6')  # Continuous

# Define the objective function
model.setObjective(3*x0**2 + 2*x0*x1 + 5*x0*x2 + 4*x1**2 + 4*x2**2 + 3*x0 + 2*x2, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(15*x0 <= 320)
model.addConstr(19*x0 <= 222)
model.addConstr(13*x0 <= 285)
model.addConstr(15*x1 <= 320)
model.addConstr(8*x1 <= 222)
model.addConstr(3*x1 <= 285)
model.addConstr(20*x2 <= 320)
model.addConstr(9*x2 <= 222)
model.addConstr(16*x2 <= 285)
model.addConstr(15*x0 + 15*x1 >= 49)
model.addConstr(15**2*x1 + 20**2*x2 >= 106)
model.addConstr(15*x0 + 15*x1 + 20*x2 >= 106)
model.addConstr(19*x0 + 8*x1 >= 66)
model.addConstr(8**2*x1 + 9**2*x2 >= 46)
model.addConstr(19*x0 + 8*x1 + 9*x2 >= 46)
model.addConstr(3*x1 + 16*x2 >= 37)
model.addConstr(13*x0 + 16*x2 >= 71)
model.addConstr(13*x0 + 3*x1 + 16*x2 >= 71)
model.addConstr(10*x0**2 - 3*x1**2 >= 0)
model.addConstr(8*x0 - 8*x2 >= 0)
model.addConstr(15*x0 + 20*x2 <= 150)
model.addConstr(15*x1 + 20*x2 <= 270)
model.addConstr(15*x0 + 15*x1 <= 320)
model.addConstr(8*x1 + 9*x2 <= 94)
model.addConstr(19*x0 + 9*x2 <= 205)
model.addConstr(13**2*x0 + 16**2*x2 <= 263)
model.addConstr(3*x1 + 16*x2 <= 121)

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', model.objval)
    print('Milligrams of zinc: ', x0.varValue)
    print('Milligrams of vitamin E: ', x1.varValue)
    print('Milligrams of vitamin B6: ', x2.varValue)
else:
    print('No solution found')
```

## Step 7: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'milligrams of zinc'),
        ('x1', 'milligrams of vitamin E'),
        ('x2', 'milligrams of vitamin B6')
    ],
    'objective_function': '3*x0^2 + 2*x0*x1 + 5*x0*x2 + 4*x1^2 + 4*x2^2 + 3*x0 + 2*x2',
    'constraints': [
        '15*x0 <= 320',
        '19*x0 <= 222',
        '13*x0 <= 285',
        '15*x1 <= 320',
        '8*x1 <= 222',
        '3*x1 <= 285',
        '20*x2 <= 320',
        '9*x2 <= 222',
        '16*x2 <= 285',
        '15*x0 + 15*x1 >= 49',
        '225*x1 + 400*x2 >= 106',
        '15*x0 + 15*x1 + 20*x2 >= 106',
        '19*x0 + 8*x1 >= 66',
        '64*x1 + 81*x2 >= 46',
        '19*x0 + 8*x1 + 9*x2 >= 46',
        '3*x1 + 16*x2 >= 37',
        '13*x0 + 16*x2 >= 71',
        '13*x0 + 3*x1 + 16*x2 >= 71',
        '10*x0^2 - 3*x1^2 >= 0',
        '8*x0 - 8*x2 >= 0',
        '15*x0 + 20*x2 <= 150',
        '15*x1 + 20*x2 <= 270',
        '15*x0 + 15*x1 <= 320',
        '8*x1 + 9*x2 <= 94',
        '19*x0 + 9*x2 <= 205',
        '169*x0 + 256*x2 <= 263',
        '3*x1 + 16*x2 <= 121'
    ]
}
```