## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- $x_1$ = tomatoes
- $x_2$ = cornichons
- $x_3$ = steaks
- $x_4$ = strawberries

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is: 
$5x_1^2 + 9x_1x_2 + 6x_1x_3 + 5x_1x_4 + 2x_2^2 + 3x_2x_3 + 2x_3^2 + 8x_3x_4 + 3x_4^2 + 2x_1 + 4x_2 + 5x_3 + 4x_4$

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $11x_1 + 17x_2 + 16x_3 + 5x_4$ is not directly a constraint but resource usage.
- $11x_1^2 + 17x_2^2 \geq 98$
- $17x_2 + 16x_3 \geq 123$
- $16x_3 + 5x_4 \geq 72$
- $17x_2 + 5x_4 \geq 114$
- $11x_1 + 16x_3 \geq 49$
- $11x_1 + 17x_2 + 16x_3 + 5x_4 \geq 49$
- $-x_1^2 + 9x_4^2 \geq 0$
- $17^2x_2^2 + 16^2x_3^2 \leq 489$
- $11x_1 + 16x_3 + 5x_4 \leq 516$
- $11^2x_1^2 + 17^2x_2^2 + 16^2x_3^2 \leq 373$
- $x_3$ is an integer.

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

# Define the model
m = gp.Model()

# Define the variables
tomatoes = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="tomatoes")
cornichons = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="cornichons")
steaks = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, type=gp.GRB.INTEGER, name="steaks")
strawberries = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="strawberries")

# Objective function
m.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, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(11*tomatoes**2 + 17*cornichons**2 >= 98, name="iron_from_tomatoes_and_cornichons")
m.addConstr(17*cornichons + 16*steaks >= 123, name="iron_from_cornichons_and_steaks")
m.addConstr(16*steaks + 5*strawberries >= 72, name="iron_from_steaks_and_strawberries")
m.addConstr(17*cornichons + 5*strawberries >= 114, name="iron_from_cornichons_and_strawberries")
m.addConstr(11*tomatoes + 16*steaks >= 49, name="iron_from_tomatoes_and_steaks")
m.addConstr(11*tomatoes + 17*cornichons + 16*steaks + 5*strawberries >= 49, name="total_iron")
m.addConstr(-tomatoes**2 + 9*strawberries**2 >= 0, name="strawberries_vs_tomatoes")
m.addConstr(17**2*cornichons**2 + 16**2*steaks**2 <= 489, name="cornichons_and_steaks_squared")
m.addConstr(11*tomatoes + 16*steaks + 5*strawberries <= 516, name="total_iron_limit")
m.addConstr(11**2*tomatoes**2 + 17**2*cornichons**2 + 16**2*steaks**2 <= 373, name="total_squared_limit")

# Optimize the model
m.optimize()

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

## 5: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'tomatoes'), ('x2', 'cornichons'), ('x3', 'steaks'), ('x4', 'strawberries')],
    'objective_function': '5*x1^2 + 9*x1*x2 + 6*x1*x3 + 5*x1*x4 + 2*x2^2 + 3*x2*x3 + 2*x3^2 + 8*x3*x4 + 3*x4^2 + 2*x1 + 4*x2 + 5*x3 + 4*x4',
    'constraints': [
        '11*x1^2 + 17*x2^2 >= 98',
        '17*x2 + 16*x3 >= 123',
        '16*x3 + 5*x4 >= 72',
        '17*x2 + 5*x4 >= 114',
        '11*x1 + 16*x3 >= 49',
        '11*x1 + 17*x2 + 16*x3 + 5*x4 >= 49',
        '-x1^2 + 9*x4^2 >= 0',
        '289*x2^2 + 256*x3^2 <= 489',
        '11*x1 + 16*x3 + 5*x4 <= 516',
        '121*x1^2 + 289*x2^2 + 256*x3^2 <= 373'
    ]
}
```