## Step 1: Define the symbolic representation of the variables
The variables are ['milligrams of vitamin B7', 'milligrams of vitamin B4', 'grams of fiber', 'grams of protein'] which can be represented symbolically as 'x1', 'x2', 'x3', 'x4' respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is: $8x_1^2 + 4x_1x_2 + 2x_1x_3 + 6x_2x_4 + 5x_3^2 + 8x_1 + 3x_3 + 4x_4$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $5x_1 \leq 213$ (cognitive performance index for $x_1$)
- $15x_1 \leq 140$ (muscle growth index for $x_1$)
- $12x_2 \leq 213$ (cognitive performance index for $x_2$)
- $17x_2 \leq 140$ (muscle growth index for $x_2$)
- $12x_3 \leq 213$ (cognitive performance index for $x_3$)
- $23x_3 \leq 140$ (muscle growth index for $x_3$)
- $11x_4 \leq 213$ (cognitive performance index for $x_4$)
- $22x_4 \leq 140$ (muscle growth index for $x_4$)
- $12x_2 + 11x_4 \geq 20$ (total combined cognitive performance index from $x_2$ and $x_4$)
- $5x_1 + 12x_3 + 11x_4 \geq 39$ (total combined cognitive performance index from $x_1$, $x_3$, and $x_4$)
- $5x_1 + 12x_2 + 12x_3 + 11x_4 \geq 39$ (total combined cognitive performance index from all variables)
- $15x_1 + 17x_2 \geq 25$ (total combined muscle growth index from $x_1$ and $x_2$)
- $17x_2 + 23x_3 \geq 12$ (total combined muscle growth index from $x_2$ and $x_3$)
- $15^2x_1^2 + 17^2x_2^2 + 22^2x_4^2 \geq 18$ (total combined muscle growth index from $x_1^2$, $x_2^2$, and $x_4^2$)
- $15x_1 + 17x_2 + 23x_3 + 22x_4 \geq 18$ (total combined muscle growth index from all variables)
- $-4x_1 + 7x_2 \geq 0$ (linear constraint)
- $15x_1 + 17x_2 + 23x_3 \leq 134$ (total combined muscle growth index from $x_1$, $x_2$, and $x_3$)

## 4: Provide the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'milligrams of vitamin B7'), ('x2', 'milligrams of vitamin B4'), ('x3', 'grams of fiber'), ('x4', 'grams of protein')],
'objective_function': '8*x1^2 + 4*x1*x2 + 2*x1*x3 + 6*x2*x4 + 5*x3^2 + 8*x1 + 3*x3 + 4*x4',
'constraints': [
'5*x1 <= 213',
'15*x1 <= 140',
'12*x2 <= 213',
'17*x2 <= 140',
'12*x3 <= 213',
'23*x3 <= 140',
'11*x4 <= 213',
'22*x4 <= 140',
'12*x2 + 11*x4 >= 20',
'5*x1 + 12*x3 + 11*x4 >= 39',
'5*x1 + 12*x2 + 12*x3 + 11*x4 >= 39',
'15*x1 + 17*x2 >= 25',
'17*x2 + 23*x3 >= 12',
'225*x1^2 + 289*x2^2 + 484*x4^2 >= 18',
'15*x1 + 17*x2 + 23*x3 + 22*x4 >= 18',
'-4*x1 + 7*x2 >= 0',
'15*x1 + 17*x2 + 23*x3 <= 134'
]
}
```

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

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

# Define the variables
x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='x1')  # milligrams of vitamin B7
x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='x2')  # milligrams of vitamin B4
x3 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='x3')  # grams of fiber
x4 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='x4')  # grams of protein

# Objective function
model.setObjective(8*x1**2 + 4*x1*x2 + 2*x1*x3 + 6*x2*x4 + 5*x3**2 + 8*x1 + 3*x3 + 4*x4, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(5*x1 <= 213)
model.addConstr(15*x1 <= 140)
model.addConstr(12*x2 <= 213)
model.addConstr(17*x2 <= 140)
model.addConstr(12*x3 <= 213)
model.addConstr(23*x3 <= 140)
model.addConstr(11*x4 <= 213)
model.addConstr(22*x4 <= 140)
model.addConstr(12*x2 + 11*x4 >= 20)
model.addConstr(5*x1 + 12*x3 + 11*x4 >= 39)
model.addConstr(5*x1 + 12*x2 + 12*x3 + 11*x4 >= 39)
model.addConstr(15*x1 + 17*x2 >= 25)
model.addConstr(17*x2 + 23*x3 >= 12)
model.addConstr(225*x1**2 + 289*x2**2 + 484*x4**2 >= 18)
model.addConstr(15*x1 + 17*x2 + 23*x3 + 22*x4 >= 18)
model.addConstr(-4*x1 + 7*x2 >= 0)
model.addConstr(15*x1 + 17*x2 + 23*x3 <= 134)

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', model.objVal)
    print('x1: ', x1.varValue)
    print('x2: ', x2.varValue)
    print('x3: ', x3.varValue)
    print('x4: ', x4.varValue)
else:
    print('The model is infeasible')
```