To solve the given optimization problem using Gurobi, we need to translate the natural language description into a mathematical formulation that can be expressed in code. The objective is to minimize the cost function \(3x_0 + 3x_1 + 5x_2\), where \(x_0\) represents cartons of milk, \(x_1\) represents packs of napkins, and \(x_2\) represents packs of paper plates.

The constraints given can be summarized as follows:
1. Cost constraints for each item: $15 per carton of milk, $8 per pack of napkins, $23 per pack of paper plates.
2. Usefulness ratings: 4 for each carton of milk, 1 for each pack of napkins, 17 for each pack of paper plates.
3. Minimum spendings: At least $35 on packs of napkins and packs of paper plates combined, at least $96 on cartons of milk plus packs of napkins.
4. Total minimum spending: At least $96 on all items combined.
5. Usefulness rating constraints:
   - Cartons of milk + packs of paper plates ≥ 84.
   - Cartons of milk + packs of napkins ≥ 58.
   - All items combined ≥ 89 (this constraint is mentioned twice, implying it's a critical requirement).
6. Additional linear constraint: \(-4x_0 + 7x_2 \geq 0\).
7. Maximum total spending: $221 on all items.
8. Integer constraints: \(x_0\), \(x_1\), and \(x_2\) must be integers.

Given these constraints, the problem can be formulated as a linear integer programming (ILP) problem. Here is how you could implement it in Gurobi using Python:

```python
from gurobipy import *

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

# Define variables
x0 = m.addVar(vtype=GRB.INTEGER, name="cartons_of_milk")
x1 = m.addVar(vtype=GRB.INTEGER, name="packs_of_napkins")
x2 = m.addVar(vtype=GRB.INTEGER, name="packs_of_paper_plates")

# Objective function: Minimize 3*x0 + 3*x1 + 5*x2
m.setObjective(3*x0 + 3*x1 + 5*x2, GRB.MINIMIZE)

# Constraints
# Minimum spending constraints
m.addConstr(8*x1 + 23*x2 >= 35, name="min_spend_napkins_paperplates")
m.addConstr(15*x0 + 8*x1 >= 96, name="min_spend_milk_napkins")
m.addConstr(15*x0 + 8*x1 + 23*x2 >= 96, name="total_min_spend")

# Usefulness rating constraints
m.addConstr(4*x0 + 17*x2 >= 84, name="usefulness_milk_paperplates")
m.addConstr(4*x0 + x1 >= 58, name="usefulness_milk_napkins")
m.addConstr(4*x0 + x1 + 17*x2 >= 89, name="total_usefulness")

# Additional linear constraint
m.addConstr(-4*x0 + 7*x2 >= 0, name="additional_linear_constraint")

# Maximum total spending constraint
m.addConstr(15*x0 + 8*x1 + 23*x2 <= 221, name="max_total_spend")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Cartons of milk: {x0.x}")
    print(f"Packs of napkins: {x1.x}")
    print(f"Packs of paper plates: {x2.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```