## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- $x_0$ represents 'milligrams of potassium'
- $x_1$ represents 'grams of carbohydrates'
- $x_2$ represents 'milligrams of calcium'
- $x_3$ represents 'milligrams of vitamin D'

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is: 
$2x_0^2 + 6x_0x_1 + 6x_0x_2 + x_0x_3 + 7x_1x_2 + 6x_1x_3 + 4x_2^2 + 4x_2x_3 + 9x_3^2 + 9x_0 + x_1 + 8x_2 + 6x_3$

## 3: Define the constraints in symbolic notation
The constraints are:
- $16x_0 \leq 248$
- $3x_0 \leq 278$
- $15x_1 \leq 248$
- $11x_1 \leq 278$
- $16x_2 \leq 248$
- $2x_2 \leq 278$
- $13x_3 \leq 248$
- $9x_3 \leq 278$
- $15x_1 + 16x_2 \geq 56$
- $16x_0 + 13x_3 \geq 48$
- $16x_0 + 15x_1 \geq 53$
- $15x_1 + 13x_3 \geq 59$
- $16x_2 + 13x_3 \geq 25$
- $16x_0 + 15x_1 + 16x_2 + 13x_3 \geq 25$
- $11x_1^2 + 9x_3^2 \geq 34$
- $3x_0 + 11x_1 \geq 38$
- $3x_0 + 11x_1 + 2x_2 + 9x_3 \geq 38$
- $15x_1 + 16x_2 \leq 142$
- $16x_2 + 13x_3 \leq 248$
- $15^2x_1^2 + 13^2x_3^2 \leq 67$
- $16x_0 + 15x_1 \leq 229$
- $16x_0 + 13x_3 \leq 205$
- $3^2x_0^2 + 11^2x_1^2 + 9^2x_3^2 \leq 132$
- $11^2x_1^2 + 2^2x_2^2 + 9^2x_3^2 \leq 188$

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

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

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

# Define the variables
x0 = model.addVar(name="milligrams_of_potassium", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x1 = model.addVar(name="grams_of_carbohydrates", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x2 = model.addVar(name="milligrams_of_calcium", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x3 = model.addVar(name="milligrams_of_vitamin_D", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

# Objective function
model.setObjective(2*x0**2 + 6*x0*x1 + 6*x0*x2 + x0*x3 + 7*x1*x2 + 6*x1*x3 + 4*x2**2 + 4*x2*x3 + 9*x3**2 + 9*x0 + x1 + 8*x2 + 6*x3, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(16*x0 <= 248)
model.addConstr(3*x0 <= 278)
model.addConstr(15*x1 <= 248)
model.addConstr(11*x1 <= 278)
model.addConstr(16*x2 <= 248)
model.addConstr(2*x2 <= 278)
model.addConstr(13*x3 <= 248)
model.addConstr(9*x3 <= 278)
model.addConstr(15*x1 + 16*x2 >= 56)
model.addConstr(16*x0 + 13*x3 >= 48)
model.addConstr(16*x0 + 15*x1 >= 53)
model.addConstr(15*x1 + 13*x3 >= 59)
model.addConstr(16*x2 + 13*x3 >= 25)
model.addConstr(16*x0 + 15*x1 + 16*x2 + 13*x3 >= 25)
model.addConstr(11*x1**2 + 9*x3**2 >= 34)
model.addConstr(3*x0 + 11*x1 >= 38)
model.addConstr(3*x0 + 11*x1 + 2*x2 + 9*x3 >= 38)
model.addConstr(15*x1 + 16*x2 <= 142)
model.addConstr(16*x2 + 13*x3 <= 248)
model.addConstr(15**2*x1**2 + 13**2*x3**2 <= 67)
model.addConstr(16*x0 + 15*x1 <= 229)
model.addConstr(16*x0 + 13*x3 <= 205)
model.addConstr(3**2*x0**2 + 11**2*x1**2 + 9**2*x3**2 <= 132)
model.addConstr(11**2*x1**2 + 2**2*x2**2 + 9**2*x3**2 <= 188)

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objVal)
    print("milligrams of potassium: ", x0.varValue)
    print("grams of carbohydrates: ", x1.varValue)
    print("milligrams of calcium: ", x2.varValue)
    print("milligrams of vitamin D: ", x3.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'milligrams of potassium'), 
        ('x1', 'grams of carbohydrates'), 
        ('x2', 'milligrams of calcium'), 
        ('x3', 'milligrams of vitamin D')
    ], 
    'objective_function': '2*x0^2 + 6*x0*x1 + 6*x0*x2 + x0*x3 + 7*x1*x2 + 6*x1*x3 + 4*x2^2 + 4*x2*x3 + 9*x3^2 + 9*x0 + x1 + 8*x2 + 6*x3', 
    'constraints': [
        '16*x0 <= 248', 
        '3*x0 <= 278', 
        '15*x1 <= 248', 
        '11*x1 <= 278', 
        '16*x2 <= 248', 
        '2*x2 <= 278', 
        '13*x3 <= 248', 
        '9*x3 <= 278', 
        '15*x1 + 16*x2 >= 56', 
        '16*x0 + 13*x3 >= 48', 
        '16*x0 + 15*x1 >= 53', 
        '15*x1 + 13*x3 >= 59', 
        '16*x2 + 13*x3 >= 25', 
        '16*x0 + 15*x1 + 16*x2 + 13*x3 >= 25', 
        '11*x1^2 + 9*x3^2 >= 34', 
        '3*x0 + 11*x1 >= 38', 
        '3*x0 + 11*x1 + 2*x2 + 9*x3 >= 38', 
        '15*x1 + 16*x2 <= 142', 
        '16*x2 + 13*x3 <= 248', 
        '15^2*x1^2 + 13^2*x3^2 <= 67', 
        '16*x0 + 15*x1 <= 229', 
        '16*x0 + 13*x3 <= 205', 
        '3^2*x0^2 + 11^2*x1^2 + 9^2*x3^2 <= 132', 
        '11^2*x1^2 + 2^2*x2^2 + 9^2*x3^2 <= 188'
    ]
}
```