## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- $x_0$ represents 'milligrams of magnesium'
- $x_1$ represents 'milligrams of vitamin B12'
- $x_2$ represents 'grams of protein'
- $x_3$ represents 'milligrams of vitamin E'

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is: 
$5x_0^2 + 5x_1x_2 + 7x_1x_3 + 8x_2x_3 + 8x_3^2 + 6x_1 + 5x_2 + 5x_3$

## Step 3: List the constraints in symbolic notation
The constraints are:
- $x_0 \geq 0$ (implicit, as it's a quantity)
- $x_1 \geq 0$ and integer
- $x_2 \geq 0$
- $x_3 \geq 0$
- $x_0 \leq 63$ (from $r_0$)
- $4x_0 + 9x_1 + 14x_2 + 6x_3 \leq 248$ (from $r_1$)
- $x_0 + 2x_1 + 5x_2 + 4x_3 \leq 63$ (total energy stability index)
- $14x_2 + 6x_3 \geq 28$ (muscle growth index from $x_2$ and $x_3$)
- $4x_0^2 + 9x_1^2 \geq 42$ (muscle growth index from $x_0^2$ and $x_1^2$)
- $9x_1^2 + 14^2x_2^2 \geq 62$ (muscle growth index from $x_1^2$ and $x_2^2$)
- $4x_0 + 9x_1 + 6x_3 \geq 59$ (muscle growth index from $x_0$, $x_1$, and $x_3$)
- $4x_0 + 14x_2 + 6x_3 \geq 59$ (muscle growth index from $x_0$, $x_2$, and $x_3$)
- $16x_0^2 + 81x_1^2 + 36x_3^2 \geq 46$ (muscle growth index from $x_0^2$, $x_1^2$, and $x_3^2$)
- $4x_0 + 14x_2 + 6x_3 \geq 46$ (muscle growth index from $x_0$, $x_2$, and $x_3$)
- $5x_2 + 4x_3 \leq 33$ (energy stability index from $x_2$ and $x_3$)
- $2x_1 + 5x_2 \leq 60$ (energy stability index from $x_1$ and $x_2$)
- $x_0^2 + 25x_2^2 \leq 52$ (energy stability index from $x_0^2$ and $x_2^2$)
- $81x_1^2 + 36x_3^2 \leq 42$ (energy stability index from $x_1^2$ and $x_3^2$)
- $x_0 + 2x_1 + 5x_2 + 4x_3 \leq 42$ (total energy stability index)
- $16x_0^2 + 81x_1^2 \leq 141$ (muscle growth index from $x_0^2$ and $x_1^2$)
- $9x_1 + 14x_2 \leq 112$ (muscle growth index from $x_1$ and $x_2$)
- $14x_2 + 6x_3 \leq 155$ (muscle growth index from $x_2$ and $x_3$)
- $16x_0^2 + 196x_2^2 \leq 69$ (muscle growth index from $x_0^2$ and $x_2^2$)
- $9x_1 + 6x_3 \leq 92$ (muscle growth index from $x_1$ and $x_3$)
- $4x_0 + 9x_1 + 14x_2 + 6x_3 \leq 92$ (muscle growth index from all)

## 4: Implement the optimization problem using Gurobi

```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(lb=0, name="milligrams of magnesium")
x1 = m.addVar(lb=0, vtype=gp.GRB.INTEGER, name="milligrams of vitamin B12")
x2 = m.addVar(lb=0, name="grams of protein")
x3 = m.addVar(lb=0, name="milligrams of vitamin E")

# Define the objective function
m.setObjective(5*x0**2 + 5*x1*x2 + 7*x1*x3 + 8*x2*x3 + 8*x3**2 + 6*x1 + 5*x2 + 5*x3, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(x0 <= 63)
m.addConstr(4*x0 + 9*x1 + 14*x2 + 6*x3 <= 248)
m.addConstr(x0 + 2*x1 + 5*x2 + 4*x3 <= 63)
m.addConstr(14*x2 + 6*x3 >= 28)
m.addConstr(4*x0**2 + 9*x1**2 >= 42)
m.addConstr(9*x1**2 + 196*x2**2 >= 62)
m.addConstr(4*x0 + 9*x1 + 6*x3 >= 59)
m.addConstr(4*x0 + 14*x2 + 6*x3 >= 59)
m.addConstr(16*x0**2 + 81*x1**2 + 36*x3**2 >= 46)
m.addConstr(4*x0 + 14*x2 + 6*x3 >= 46)
m.addConstr(5*x2 + 4*x3 <= 33)
m.addConstr(2*x1 + 5*x2 <= 60)
m.addConstr(x0**2 + 25*x2**2 <= 52)
m.addConstr(81*x1**2 + 36*x3**2 <= 42)
m.addConstr(x0 + 2*x1 + 5*x2 + 4*x3 <= 42)
m.addConstr(16*x0**2 + 81*x1**2 <= 141)
m.addConstr(9*x1 + 14*x2 <= 112)
m.addConstr(14*x2 + 6*x3 <= 155)
m.addConstr(16*x0**2 + 196*x2**2 <= 69)
m.addConstr(9*x1 + 6*x3 <= 92)
m.addConstr(4*x0 + 9*x1 + 14*x2 + 6*x3 <= 92)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print("milligrams of magnesium:", x0.varValue)
    print("milligrams of vitamin B12:", x1.varValue)
    print("grams of protein:", x2.varValue)
    print("milligrams of vitamin E:", x3.varValue)
else:
    print("No optimal solution found.")
```

## 5: Symbolic representation of the problem

```json
{
    'sym_variables': [
        ('x0', 'milligrams of magnesium'), 
        ('x1', 'milligrams of vitamin B12'), 
        ('x2', 'grams of protein'), 
        ('x3', 'milligrams of vitamin E')
    ], 
    'objective_function': '5*x0^2 + 5*x1*x2 + 7*x1*x3 + 8*x2*x3 + 8*x3^2 + 6*x1 + 5*x2 + 5*x3', 
    'constraints': [
        'x0 <= 63', 
        '4*x0 + 9*x1 + 14*x2 + 6*x3 <= 248', 
        'x0 + 2*x1 + 5*x2 + 4*x3 <= 63', 
        '14*x2 + 6*x3 >= 28', 
        '4*x0^2 + 9*x1^2 >= 42', 
        '9*x1^2 + 196*x2^2 >= 62', 
        '4*x0 + 9*x1 + 6*x3 >= 59', 
        '4*x0 + 14*x2 + 6*x3 >= 59', 
        '16*x0^2 + 81*x1^2 + 36*x3^2 >= 46', 
        '4*x0 + 14*x2 + 6*x3 >= 46', 
        '5*x2 + 4*x3 <= 33', 
        '2*x1 + 5*x2 <= 60', 
        'x0^2 + 25*x2^2 <= 52', 
        '81*x1^2 + 36*x3^2 <= 42', 
        'x0 + 2*x1 + 5*x2 + 4*x3 <= 42', 
        '16*x0^2 + 81*x1^2 <= 141', 
        '9*x1 + 14*x2 <= 112', 
        '14*x2 + 6*x3 <= 155', 
        '16*x0^2 + 196*x2^2 <= 69', 
        '9*x1 + 6*x3 <= 92', 
        '4*x0 + 9*x1 + 14*x2 + 6*x3 <= 92'
    ]
}
```