## Step 1: Define the optimization problem and identify the variables, objective function, and constraints.

The optimization problem involves minimizing an objective function with variables representing the quantities of tomatoes, cornichons, steaks, and strawberries. The objective function to minimize is:
\[ 5 \times \text{tomatoes}^2 + 9 \times \text{tomatoes} \times \text{cornichons} + 6 \times \text{tomatoes} \times \text{steaks} + 5 \times \text{tomatoes} \times \text{strawberries} + 2 \times \text{cornichons}^2 + 3 \times \text{cornichons} \times \text{steaks} + 2 \times \text{steaks}^2 + 8 \times \text{steaks} \times \text{strawberries} + 3 \times \text{strawberries}^2 + 2 \times \text{tomatoes} + 4 \times \text{cornichons} + 5 \times \text{steaks} + 4 \times \text{strawberries} \]

The variables are:
- $x_0 = \text{tomatoes}$
- $x_1 = \text{cornichons}$
- $x_2 = \text{steaks}$
- $x_3 = \text{strawberries}$

The constraints are given based on the iron content and other conditions.

## 2: Translate the constraints into mathematical expressions.

1. Iron content in each item: 
   - Tomatoes: $11$ mg
   - Cornichons: $17$ mg
   - Steaks: $16$ mg
   - Strawberries: $5$ mg

2. Constraints:
   - $11^2 \times x_0^2 + 17^2 \times x_1^2 \geq 98$
   - $17 \times x_1 + 16 \times x_2 \geq 123$
   - $16 \times x_2 + 5 \times x_3 \geq 72$
   - $17 \times x_1 + 5 \times x_3 \geq 114$
   - $11 \times x_0 + 16 \times x_2 \geq 49$
   - $11 \times x_0 + 17 \times x_1 + 16 \times x_2 + 5 \times x_3 \geq 49$
   - $-1 \times x_0^2 + 9 \times x_3^2 \geq 0$
   - $17^2 \times x_1^2 + 16^2 \times x_2^2 \leq 489$
   - $11 \times x_0 + 16 \times x_2 + 5 \times x_3 \leq 516$
   - $11^2 \times x_0^2 + 17^2 \times x_1^2 + 16^2 \times x_2^2 \leq 373$

## 3: Implement the problem in Gurobi.

```python
import gurobi

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

    # Define variables
    tomatoes = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="tomatoes")
    cornichons = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="cornichons")
    steaks = model.addVar(lb=0, ub=gurobi.GRB.INFINITY, integrality=gurobi.GRB.INTEGER, name="steaks")
    strawberries = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="strawberries")

    # Objective function
    model.setObjective(5 * tomatoes**2 + 9 * tomatoes * cornichons + 6 * tomatoes * steaks + 5 * tomatoes * strawberries +
                       2 * cornichons**2 + 3 * cornichons * steaks + 2 * steaks**2 + 8 * steaks * strawberries +
                       3 * strawberries**2 + 2 * tomatoes + 4 * cornichons + 5 * steaks + 4 * strawberries,
                       gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(121 * tomatoes**2 + 289 * cornichons**2 >= 98, name="iron_tomatoes_cornichons")
    model.addConstr(17 * cornichons + 16 * steaks >= 123, name="iron_cornichons_steaks")
    model.addConstr(16 * steaks + 5 * strawberries >= 72, name="iron_steaks_strawberries")
    model.addConstr(17 * cornichons + 5 * strawberries >= 114, name="iron_cornichons_strawberries")
    model.addConstr(11 * tomatoes + 16 * steaks >= 49, name="iron_tomatoes_steaks")
    model.addConstr(11 * tomatoes + 17 * cornichons + 16 * steaks + 5 * strawberries >= 49, name="iron_all")
    model.addConstr(-tomatoes**2 + 9 * strawberries**2 >= 0, name="strawberries_vs_tomatoes")
    model.addConstr(289 * cornichons**2 + 256 * steaks**2 <= 489, name="iron_cornichons_steaks_limit")
    model.addConstr(11 * tomatoes + 16 * steaks + 5 * strawberries <= 516, name="iron_tomatoes_steaks_strawberries_limit")
    model.addConstr(121 * tomatoes**2 + 289 * cornichons**2 + 256 * steaks**2 <= 373, name="iron_all_limit")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Tomatoes: ", tomatoes.varValue)
        print("Cornichons: ", cornichons.varValue)
        print("Steaks: ", steaks.varValue)
        print("Strawberries: ", strawberries.varValue)
    else:
        print("Model is infeasible")

optimize_problem()
```