## Step 1: Define the optimization problem
The problem is to minimize the given objective function subject to several constraints. The objective function is:
\[ 1.17x_0^2 + 6.23x_0x_1 + 3.65x_0x_2 + 3.75x_1^2 + 1.87x_1x_2 + 6.35x_0 + 8.12x_1 + 2.37x_2 \]
where $x_0$ represents the quantity of fruit salads, $x_1$ represents the quantity of sashimi, and $x_2$ represents the quantity of bowls of instant ramen.

## 2: List all constraints
The constraints are:
1. $6.94x_0 + 3.98x_1 + 7.82x_2 \leq 133$ (grams of fat)
2. $5.74x_0 + 6.87x_1 + 1.66x_2 \leq 182$ (milligrams of iron)
3. $3.98x_1 + 7.82x_2 \geq 27$ (at least 27 grams of fat from sashimi and bowls of instant ramen)
4. $6.94x_0 + 7.82x_2 \geq 44$ (at least 44 grams of fat from fruit salads and bowls of instant ramen)
5. $6.94x_0 + 3.98x_1 + 7.82x_2 \geq 44$ (at least 44 grams of fat from all sources)
6. $5.74x_0 + 1.66x_2 \geq 20$ (at least 20 milligrams of iron from fruit salads and bowls of instant ramen)
7. $5.74x_0 + 6.87x_1 + 1.66x_2 \geq 40$ (at least 40 milligrams of iron from all sources)
8. $-10x_1 + 7x_2 \geq 0$
9. $-x_0 + 5x_1 \geq 0$
10. $6.94^2x_0^2 + 3.98^2x_1^2 + 7.82^2x_2^2 \leq 65^2$ (at most 65 grams of fat squared)
11. $5.74^2x_0^2 + 6.87^2x_1^2 + 1.66^2x_2^2 \leq 122^2$ (at most 122 milligrams of iron squared)
12. $x_0$ and $x_1$ are integers, $x_2$ can be a real number.

## 3: Convert the problem into Gurobi code
```python
import gurobi

def optimize_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="fruit_salads", vtype=gurobi.GRB.INTEGER)  # integer
    x1 = model.addVar(name="sashimi", vtype=gurobi.GRB.INTEGER)  # integer
    x2 = model.addVar(name="bowls_of_instant_ramen")  # continuous

    # Objective function
    model.setObjective(1.17*x0**2 + 6.23*x0*x1 + 3.65*x0*x2 + 3.75*x1**2 + 1.87*x1*x2 + 6.35*x0 + 8.12*x1 + 2.37*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(6.94*x0 + 3.98*x1 + 7.82*x2 <= 133)  # grams of fat
    model.addConstr(5.74*x0 + 6.87*x1 + 1.66*x2 <= 182)  # milligrams of iron
    model.addConstr(3.98*x1 + 7.82*x2 >= 27)  # at least 27 grams of fat from sashimi and bowls of instant ramen
    model.addConstr(6.94*x0 + 7.82*x2 >= 44)  # at least 44 grams of fat from fruit salads and bowls of instant ramen
    model.addConstr(6.94*x0 + 3.98*x1 + 7.82*x2 >= 44)  # at least 44 grams of fat from all sources
    model.addConstr(5.74*x0 + 1.66*x2 >= 20)  # at least 20 milligrams of iron from fruit salads and bowls of instant ramen
    model.addConstr(5.74*x0 + 6.87*x1 + 1.66*x2 >= 40)  # at least 40 milligrams of iron from all sources
    model.addConstr(-10*x1 + 7*x2 >= 0)
    model.addConstr(-x0 + 5*x1 >= 0)
    model.addConstr(6.94**2*x0**2 + 3.98**2*x1**2 + 7.82**2*x2**2 <= 65**2)  # at most 65 grams of fat squared
    model.addConstr(5.74**2*x0**2 + 6.87**2*x1**2 + 1.66**2*x2**2 <= 122**2)  # at most 122 milligrams of iron squared

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Fruit Salads: ", x0.varValue)
        print("Sashimi: ", x1.varValue)
        print("Bowls of Instant Ramen: ", x2.varValue)
    else:
        print("The model is infeasible")

optimize_problem()
```