To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the provided information.

Let's denote:
- $x_1$ as the amount of pickles,
- $x_2$ as the quantity of oranges.

The objective function described is to minimize: 
\[4(x_1)^2 + 6(x_1)(x_2) + (x_2)^2 + 6(x_2)\]

Given constraints:
1. Umami index constraint: $6x_1 + 9x_2 \geq 17$
2. Cost constraint: $x_1 + 5x_2 \geq 34$
3. Additional linear constraint: $-5x_1 + 3x_2 \geq 0$
4. Squared umami index constraint: $(6x_1)^2 + (9x_2)^2 \leq 38$
5. Squared cost constraint: $(x_1)^2 + (5x_2)^2 \leq 39$

The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'pickles'), ('x2', 'oranges')],
    'objective_function': '4*(x1)**2 + 6*(x1)*(x2) + (x2)**2 + 6*(x2)',
    'constraints': [
        '6*x1 + 9*x2 >= 17',
        'x1 + 5*x2 >= 34',
        '-5*x1 + 3*x2 >= 0',
        '(6*x1)**2 + (9*x2)**2 <= 38',
        '(x1)**2 + (5*x2)**2 <= 39'
    ]
}
```

Now, let's implement this problem using Gurobi in Python:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="pickles")
x2 = m.addVar(vtype=GRB.INTEGER, name="oranges")

# Set the objective function
m.setObjective(4*(x1)**2 + 6*(x1)*(x2) + (x2)**2 + 6*(x2), GRB.MINIMIZE)

# Add constraints
m.addConstr(6*x1 + 9*x2 >= 17, name="umami_index_constraint")
m.addConstr(x1 + 5*x2 >= 34, name="cost_constraint")
m.addConstr(-5*x1 + 3*x2 >= 0, name="additional_linear_constraint")
m.addConstr((6*x1)**2 + (9*x2)**2 <= 38, name="squared_umami_index_constraint")
m.addConstr((x1)**2 + (5*x2)**2 <= 39, name="squared_cost_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Pickles: {x1.x}")
    print(f"Oranges: {x2.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```