To solve this optimization problem using Gurobi, we first need to formulate the problem in a way that Gurobi can understand. This involves defining the variables, the objective function, and all the constraints.

Let's denote:
- \(x_0\) as milligrams of vitamin B4,
- \(x_1\) as milligrams of vitamin B3,
- \(x_2\) as grams of protein,
- \(x_3\) as milligrams of vitamin C.

## Step 1: Define the Objective Function
The objective function to minimize is:
\[9x_0^2 + 6x_0x_1 + 2x_0x_2 + 4x_0x_3 + 8x_1^2 + 9x_1x_2 + 4x_1x_3 + 3x_2^2 + 8x_2x_3 + 8x_3^2 + 8x_0 + 5x_1 + 7x_2 + 9x_3\]

## Step 2: Define the Constraints
### Resource Constraints
Given:
- \(r0 = \{x0: 27, x1: 11, x2: 15, x3: 18\}\)
- \(r1 = \{x0: 1, x1: 26, x2: 15, x3: 2\}\)

### Linear Constraints
1. \(27x_0 + 11x_1 \geq 19\)
2. \(11x_1 + 15x_2 \geq 24\)
3. \(11x_1 + 18x_3 \geq 29\)
4. \(27x_0 + 15x_2 \geq 33\)
5. \(27x_0 + 15x_2 + 18x_3 \geq 56\)
6. \(27x_0 + 11x_1 + 15x_2 \geq 56\)
7. \(27x_0 + 15x_2 + 18x_3 \geq 54\)
8. \(27x_0 + 11x_1 + 15x_2 \geq 54\)
9. \(27x_0 + 11x_1 + 15x_2 + 18x_3 \geq 54\)
10. \(x_0 + 15x_2 \geq 38\)
11. \(15x_2 + 2x_3 \geq 56\)
12. \(x_0 + 26x_1 + 2x_3 \geq 53\)
13. \(x_0 + 26x_1 + 15x_2 + 2x_3 \geq 53\)
14. \(7x_2^2 - 4x_3^2 \geq 0\)
15. \(-10x_0 + 9x_3 \geq 0\)
16. \(15x_2^2 + 18x_3^2 \leq 108\)
17. \(27x_0 + 18x_3 \leq 207\)
18. \(11x_1 + 15x_2 \leq 121\)
19. \(27x_0 + 15x_2 \leq 118\)
20. \(27x_0 + 15x_2 + 18x_3 \leq 80\)
21. \(27x_0 + 11x_1 + 18x_3 \leq 151\)

### Variable Constraints
- \(x_0\) is an integer
- \(x_1\) is an integer
- \(x_2\) is an integer
- \(x_3\) is continuous

## Step 3: Implement in Gurobi
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()
    
    # Define variables
    x0 = model.addVar(name="milligrams_of_vitamin_B4", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="milligrams_of_vitamin_B3", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="grams_of_protein", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="milligrams_of_vitamin_C")
    
    # Objective function
    model.setObjective(9*x0**2 + 6*x0*x1 + 2*x0*x2 + 4*x0*x3 + 8*x1**2 + 9*x1*x2 + 4*x1*x3 + 
                       3*x2**2 + 8*x2*x3 + 8*x3**2 + 8*x0 + 5*x1 + 7*x2 + 9*x3, gurobi.GRB.MINIMIZE)
    
    # Constraints
    model.addConstr(27*x0 + 11*x1 >= 19)
    model.addConstr(11*x1 + 15*x2 >= 24)
    model.addConstr(11*x1 + 18*x3 >= 29)
    model.addConstr(27*x0 + 15*x2 >= 33)
    model.addConstr(27*x0 + 15*x2 + 18*x3 >= 56)
    model.addConstr(27*x0 + 11*x1 + 15*x2 >= 56)
    model.addConstr(27*x0 + 15*x2 + 18*x3 >= 54)
    model.addConstr(27*x0 + 11*x1 + 15*x2 >= 54)
    model.addConstr(27*x0 + 11*x1 + 15*x2 + 18*x3 >= 54)
    model.addConstr(x0 + 15*x2 >= 38)
    model.addConstr(15*x2 + 2*x3 >= 56)
    model.addConstr(x0 + 26*x1 + 2*x3 >= 53)
    model.addConstr(x0 + 26*x1 + 15*x2 + 2*x3 >= 53)
    model.addConstr(7*x2**2 - 4*x3**2 >= 0)
    model.addConstr(-10*x0 + 9*x3 >= 0)
    model.addConstr(15*x2**2 + 18*x3**2 <= 108)
    model.addConstr(27*x0 + 18*x3 <= 207)
    model.addConstr(11*x1 + 15*x2 <= 121)
    model.addConstr(27*x0 + 15*x2 <= 118)
    model.addConstr(27*x0 + 15*x2 + 18*x3 <= 80)
    model.addConstr(27*x0 + 11*x1 + 18*x3 <= 151)
    
    # Optimize
    model.optimize()
    
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin B4: {x0.varValue}")
        print(f"Milligrams of vitamin B3: {x1.varValue}")
        print(f"Grams of protein: {x2.varValue}")
        print(f"Milligrams of vitamin C: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```