## Step 1: Define the symbolic representation of the variables
The variables are 'milligrams of vitamin D', 'milligrams of calcium', and 'milligrams of vitamin B7'. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $3x_0^2 + 6x_0x_1 + 8x_1^2 + 8x_0 + 3x_1$.

## 3: Define the constraints in symbolic notation
- The digestive support index of $x_0$ is 5: $5x_0 \leq 101$ is not correct, it is an equality $5 = 5$ which does not limit $x_0$, but $x_0$ has a digestive support index of 5.
- The energy stability index of $x_0$ is 15: $15x_0 \leq 97$.
- The cardiovascular support index of $x_0$ is 4: $4x_0 \leq 320$.
- The digestive support index of $x_1$ is 20: $20x_1 \leq 101$.
- The energy stability index of $x_1$ is 3: $3x_1 \leq 97$.
- The cardiovascular support index of $x_1$ is 8: $8x_1 \leq 320$.
- The digestive support index of $x_2$ is 19: $19x_2 \leq 101$.
- The energy stability index of $x_2$ is 9: $9x_2 \leq 97$.
- The cardiovascular support index of $x_2$ is 7: $7x_2 \leq 320$.
- $x_0^2 + x_1^2 + x_2^2 \geq 84$.
- $20x_1 + 19x_2 \leq 62$.
- $5x_0 + 19x_2 \leq 48$.
- $5x_0 + 20x_1 + 19x_2 \leq 48$.
- $15x_0 + 9x_2 \leq 76$.
- $15^2x_0^2 + 3^2x_1^2 \leq 42^2$ or $(15x_0)^2 + (3x_1)^2 \leq 42^2$ which simplifies to $225x_0^2 + 9x_1^2 \leq 1764$.
- $15x_0 + 3x_1 + 9x_2 \leq 42$.
- $8^2x_1^2 + 7^2x_2^2 \leq 131^2$ or $64x_1^2 + 49x_2^2 \leq 17161$.
- $4^2x_0^2 + 7^2x_2^2 \leq 175^2$ or $16x_0^2 + 49x_2^2 \leq 30625$.
- $4x_0 + 8x_1 + 7x_2 \leq 124$.

## 4: Write down the symbolic problem representation
```json
{
    'sym_variables': [('x0', 'milligrams of vitamin D'), ('x1', 'milligrams of calcium'), ('x2', 'milligrams of vitamin B7')],
    'objective_function': '3*x0^2 + 6*x0*x1 + 8*x1^2 + 8*x0 + 3*x1',
    'constraints': [
        '15*x0 <= 97',
        '4*x0 <= 320',
        '3*x1 <= 97',
        '8*x1 <= 320',
        '9*x2 <= 97',
        '7*x2 <= 320',
        'x0^2 + x1^2 + x2^2 >= 84',
        '20*x1 + 19*x2 <= 62',
        '5*x0 + 19*x2 <= 48',
        '5*x0 + 20*x1 + 19*x2 <= 48',
        '15*x0 + 9*x2 <= 76',
        '225*x0^2 + 9*x1^2 <= 1764',
        '15*x0 + 3*x1 + 9*x2 <= 42',
        '64*x1^2 + 49*x2^2 <= 17161',
        '16*x0^2 + 49*x2^2 <= 30625',
        '4*x0 + 8*x1 + 7*x2 <= 124'
    ]
}
```

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

def optimize_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, name="x0")  # milligrams of vitamin D
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, name="x1")  # milligrams of calcium
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, name="x2")  # milligrams of vitamin B7

    # Objective function
    model.setObjective(3*x0**2 + 6*x0*x1 + 8*x1**2 + 8*x0 + 3*x1, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(15*x0 <= 97)
    model.addConstr(4*x0 <= 320)
    model.addConstr(3*x1 <= 97)
    model.addConstr(8*x1 <= 320)
    model.addConstr(9*x2 <= 97)
    model.addConstr(7*x2 <= 320)
    model.addConstr(x0**2 + x1**2 + x2**2 >= 84)
    model.addConstr(20*x1 + 19*x2 <= 62)
    model.addConstr(5*x0 + 19*x2 <= 48)
    model.addConstr(5*x0 + 20*x1 + 19*x2 <= 48)
    model.addConstr(15*x0 + 9*x2 <= 76)
    model.addConstr(225*x0**2 + 9*x1**2 <= 1764)
    model.addConstr(15*x0 + 3*x1 + 9*x2 <= 42)
    model.addConstr(64*x1**2 + 49*x2**2 <= 17161)
    model.addConstr(16*x0**2 + 49*x2**2 <= 30625)
    model.addConstr(4*x0 + 8*x1 + 7*x2 <= 124)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin D: {x0.varValue}")
        print(f"Milligrams of calcium: {x1.varValue}")
        print(f"Milligrams of vitamin B7: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```