To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the objective function and constraints provided.

The objective function to maximize is:
\[1 \times (\text{milligrams of iron})^2 + 3 \times (\text{milligrams of iron} \times \text{grams of fiber}) + 4 \times (\text{milligrams of iron} \times \text{milligrams of vitamin A}) + 9 \times (\text{grams of fiber} \times \text{milligrams of vitamin A}) + 7 \times (\text{milligrams of vitamin A})^2 + 3 \times (\text{grams of fiber})\]

Given constraints:
1. Cognitive performance index for milligrams of iron: 27.
2. Cognitive performance index for grams of fiber: 14.
3. Cognitive performance index for milligrams of vitamin A: 1.
4. Minimum combined cognitive performance index from milligrams of iron and milligrams of vitamin A: 58.
5. Minimum total combined cognitive performance index from all three: 62.
6. Maximum combined cognitive performance index from the squares of milligrams of iron and milligrams of vitamin A: 177.
7. Maximum total combined cognitive performance index from all three: 177.

Let's denote:
- \(x_0\) as the amount of milligrams of iron,
- \(x_1\) as the grams of fiber, which must be an integer,
- \(x_2\) as the milligrams of vitamin A.

Thus, we have:
- Objective function: Maximize \(x_0^2 + 3x_0x_1 + 4x_0x_2 + 9x_1x_2 + 7x_2^2 + 3x_1\)
- Constraints:
  - \(27x_0 + x_2 \geq 58\) (Minimum combined index for iron and vitamin A),
  - \(27x_0 + 14x_1 + x_2 \geq 62\) (Minimum total combined index),
  - \(x_0^2 + x_2^2 \leq 177\) (Maximum from squares of iron and vitamin A),
  - \(27x_0 + 14x_1 + x_2 \leq 177\) (Maximum total combined index),
  - \(x_1\) is an integer.

Here's how you could model this problem using Gurobi in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Create variables
x0 = m.addVar(lb=0, name="milligrams_of_iron")  # Amount of milligrams of iron
x1 = m.addVar(vtype=GRB.INTEGER, lb=0, name="grams_of_fiber")  # Grams of fiber, integer
x2 = m.addVar(lb=0, name="milligrams_of_vitamin_A")  # Milligrams of vitamin A

# Objective function: Maximize
m.setObjective(x0**2 + 3*x0*x1 + 4*x0*x2 + 9*x1*x2 + 7*x2**2 + 3*x1, GRB.MAXIMIZE)

# Constraints
m.addConstr(27*x0 + x2 >= 58, name="min_combined_index_iron_vitaminA")
m.addConstr(27*x0 + 14*x1 + x2 >= 62, name="min_total_combined_index")
m.addConstr(x0**2 + x2**2 <= 177, name="max_from_squares_iron_vitaminA")
m.addConstr(27*x0 + 14*x1 + x2 <= 177, name="max_total_combined_index")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"milligrams_of_iron: {x0.x}")
    print(f"grams_of_fiber: {x1.x}")
    print(f"milligrams_of_vitamin_A: {x2.x}")
else:
    print("No optimal solution found.")
```