## Step 1: Define the optimization problem
The problem is to maximize the given objective function subject to various constraints. The objective function and constraints involve variables representing grams of fiber, grams of carbohydrates, milligrams of iron, milligrams of vitamin B2, and milligrams of potassium.

## Step 2: Identify the variables and their attributes
Let $x_0$ be grams of fiber, $x_1$ be grams of carbohydrates, $x_2$ be milligrams of iron, $x_3$ be milligrams of vitamin B2, and $x_4$ be milligrams of potassium.

## 3: Formulate the objective function
The objective function to maximize is:
\[ 8x_0^2 + 4x_0x_1 + x_0x_3 + 7x_0x_4 + 6x_1^2 + 6x_1x_3 + 7x_1x_4 + 4x_2^2 + 7x_2x_3 + x_2x_4 + 4x_3x_4 + 3x_4^2 + 8x_0 + 7x_2 + 3x_4 \]

## 4: Define the constraints
Constraints from the problem statement:
- $x_0 \geq 0$ (implicit, as quantities cannot be negative)
- $x_1 \geq 0$ (implicit)
- $x_2 \geq 0$ (implicit)
- $x_3 \geq 0$ (implicit)
- $x_4 \geq 0$ (implicit)
- $x_2$ is an integer
- The immune support index constraints:
  - $1x_0 + 11x_1 + 15x_2 + 22x_3 + 5x_4 \leq 263$
  - $11x_1 + 15x_2 \geq 33$
  - $15x_2^2 + 22x_3^2 \geq 42$
  - $1x_0 + 11x_1 \geq 43$
  - $11x_3 + 5x_4 \leq 147$
  - $15x_2^2 + 5x_4^2 \leq 142$
  - $1x_0^2 + 5x_4^2 \leq 258$
  - $1x_0^2 + 22x_3^2 \leq 97$
  - $11x_1 + 5x_4 \leq 131$
  - $1x_0 + 22x_3 + 5x_4 \leq 67$
  - $1x_0 + 11x_1 + 5x_4 \leq 56$
  - $15x_2^2 + 22x_3^2 + 5x_4^2 \leq 205$
  - $1x_0 + 11x_1 + 15x_2 + 22x_3 + 5x_4 \leq 205$
- The energy stability index constraints:
  - $14x_0 + 18x_1 + 4x_2 + 11x_3 + 20x_4 \leq 332$
  - $14x_0 + 4x_2 \geq 61$
  - $14x_0 + 11x_3 \geq 34$
  - $14x_0^2 + 20x_4^2 \geq 25$
  - $4x_2 + 20x_4 \geq 49$
  - $18x_1 + 11x_3 \geq 60$
  - $-9x_0^2 + 9x_1^2 + 5x_4^2 \geq 0$
  - $4x_2 + 11x_3 \leq 160$
  - $14x_0 + 4x_2 \leq 201$
  - $11x_3^2 + 20x_4^2 \leq 108$
  - $18x_1 + 4x_2 + 20x_4 \leq 322$
  - $14x_0 + 18x_1 + 11x_3 \leq 67$
  - $14x_0 + 18x_1 + 4x_2 + 11x_3 + 20x_4 \leq 67$

## 5: Implement the problem in Gurobi
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(lb=0, name="grams_of_fiber")  # Continuous
x1 = m.addVar(lb=0, name="grams_of_carbohydrates")  # Continuous
x2 = m.addVar(lb=0, type=gp.GRB.INTEGER, name="milligrams_of_iron")  # Integer
x3 = m.addVar(lb=0, name="milligrams_of_vitamin_B2")  # Continuous
x4 = m.addVar(lb=0, name="milligrams_of_potassium")  # Continuous

# Objective function
m.setObjective(8*x0**2 + 4*x0*x1 + x0*x3 + 7*x0*x4 + 6*x1**2 + 6*x1*x3 + 7*x1*x4 + 
               4*x2**2 + 7*x2*x3 + x2*x4 + 4*x3*x4 + 3*x4**2 + 8*x0 + 7*x2 + 3*x4, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(1*x0 + 11*x1 + 15*x2 + 22*x3 + 5*x4 <= 263, "immune_support_index")
m.addConstr(11*x1 + 15*x2 >= 33, "immune_support_index_2")
m.addConstr(15*x2**2 + 22*x3**2 >= 42, "immune_support_index_3")
m.addConstr(1*x0 + 11*x1 >= 43, "immune_support_index_4")
m.addConstr(14*x0 + 18*x1 + 4*x2 + 11*x3 + 20*x4 <= 332, "energy_stability_index")
m.addConstr(14*x0 + 4*x2 >= 61, "energy_stability_index_2")
m.addConstr(14*x0 + 11*x3 >= 34, "energy_stability_index_3")
m.addConstr(14*x0**2 + 20*x4**2 >= 25, "energy_stability_index_4")
m.addConstr(4*x2 + 20*x4 >= 49, "energy_stability_index_5")
m.addConstr(18*x1 + 11*x3 >= 60, "energy_stability_index_6")
m.addConstr(-9*x0**2 + 9*x1**2 + 5*x4**2 >= 0, "energy_stability_index_7")
m.addConstr(11*x3 + 5*x4 <= 147, "immune_support_index_8")
m.addConstr(15*x2**2 + 5*x4**2 <= 142, "immune_support_index_9")
m.addConstr(1*x0**2 + 5*x4**2 <= 258, "immune_support_index_10")
m.addConstr(1*x0**2 + 22*x3**2 <= 97, "immune_support_index_11")
m.addConstr(11*x1 + 5*x4 <= 131, "immune_support_index_12")
m.addConstr(1*x0 + 22*x3 + 5*x4 <= 67, "immune_support_index_13")
m.addConstr(1*x0 + 11*x1 + 5*x4 <= 56, "immune_support_index_14")
m.addConstr(15*x2**2 + 22*x3**2 + 5*x4**2 <= 205, "immune_support_index_15")
m.addConstr(1*x0 + 11*x1 + 15*x2 + 22*x3 + 5*x4 <= 205, "immune_support_index_16")
m.addConstr(4*x2 + 11*x3 <= 160, "energy_stability_index_8")
m.addConstr(14*x0 + 4*x2 <= 201, "energy_stability_index_9")
m.addConstr(11*x3**2 + 20*x4**2 <= 108, "energy_stability_index_10")
m.addConstr(18*x1 + 4*x2 + 20*x4 <= 322, "energy_stability_index_11")
m.addConstr(14*x0 + 18*x1 + 11*x3 <= 67, "energy_stability_index_12")
m.addConstr(14*x0 + 18*x1 + 4*x2 + 11*x3 + 20*x4 <= 67, "energy_stability_index_13")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Grams of fiber: ", x0.varValue)
    print("Grams of carbohydrates: ", x1.varValue)
    print("Milligrams of iron: ", x2.varValue)
    print("Milligrams of vitamin B2: ", x3.varValue)
    print("Milligrams of potassium: ", x4.varValue)
else:
    print("The model is infeasible or unbounded.")
```