To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the constraints and objective function provided. The goal is to minimize the value of \(5x_0^2 + 7x_1^2 + x_1\), where \(x_0\) represents the quantity of protein bars and \(x_1\) represents the quantity of hot dogs.

Given constraints:
- Fiber from protein bars squared plus hot dogs squared should be at least 26 grams.
- Fiber from protein bars plus hot dogs should be at least 26 grams.
- Total tastiness rating from protein bars squared plus hot dogs squared should be at least 38.
- Total tastiness rating from protein bars plus hot dogs should be at least 38.
- Total umami index from protein bars plus hot dogs should be at least 27.
- \(x_0 - 9x_1 \geq 0\).
- Fiber from protein bars and hot dogs should not exceed 66 grams.
- Total tastiness rating from protein bars plus hot dogs should not exceed 66.
- Total umami index from protein bars and hot dogs should not exceed 75.

Let's formulate these constraints in terms of \(x_0\) (protein bars) and \(x_1\) (hot dogs):
- \(16x_0^2 + 16x_1^2 \geq 26\)
- \(16x_0 + 16x_1 \geq 26\)
- \(11x_0^2 + x_1^2 \geq 38\)
- \(11x_0 + x_1 \geq 38\)
- \(15x_0 + 15x_1 \geq 27\)
- \(x_0 - 9x_1 \geq 0\)
- \(16x_0 + 16x_1 \leq 66\)
- \(11x_0 + x_1 \leq 66\)
- \(15x_0 + 15x_1 \leq 75\)

Now, let's write the Gurobi code to solve this optimization problem:

```python
from gurobipy import *

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

# Define variables
x0 = m.addVar(vtype=GRB.CONTINUOUS, name="protein_bars")
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hot_dogs")

# Objective function: minimize 5*x0^2 + 7*x1^2 + x1
m.setObjective(5*x0**2 + 7*x1**2 + x1, GRB.MINIMIZE)

# Constraints
m.addConstr(16*x0**2 + 16*x1**2 >= 26, "fiber_squared")
m.addConstr(16*x0 + 16*x1 >= 26, "fiber_linear")
m.addConstr(11*x0**2 + x1**2 >= 38, "tastiness_squared")
m.addConstr(11*x0 + x1 >= 38, "tastiness_linear")
m.addConstr(15*x0 + 15*x1 >= 27, "umami_index")
m.addConstr(x0 - 9*x1 >= 0, "protein_hotdog_ratio")
m.addConstr(16*x0 + 16*x1 <= 66, "fiber_upper_bound")
m.addConstr(11*x0 + x1 <= 66, "tastiness_upper_bound")
m.addConstr(15*x0 + 15*x1 <= 75, "umami_index_upper_bound")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Protein bars: {x0.x}")
    print(f"Hot dogs: {x1.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```