To solve the optimization problem described, we will use the Gurobi Python interface. The problem involves minimizing a quadratic objective function subject to several linear and quadratic constraints. 

First, let's clarify the objective function and the constraints:

- Objective Function: Minimize \(5x_0^2 + 4x_0x_1 + 3x_1^2 + 8x_0 + 7x_1\), where \(x_0\) represents milligrams of iron and \(x_1\) represents milligrams of potassium.

- Constraints:
  1. Immune support index for iron: \(2x_0\)
  2. Muscle growth index for iron: \(10x_0\)
  3. Immune support index for potassium: \(9x_1\)
  4. Muscle growth index for potassium: \(3x_1\)
  5. Total combined immune support index (squared terms): \(2^2x_0^2 + 9^2x_1^2 \geq 48\)
  6. Total combined immune support index: \(2x_0 + 9x_1 \geq 48\)
  7. Total combined muscle growth index: \(10x_0 + 3x_1 \geq 18\)
  8. Another formulation of total combined muscle growth index (same as above): \(10x_0 + 3x_1 \geq 18\)
  9. Linear constraint: \(-6x_0 + 5x_1 \geq 0\)
  10. Upper bound on total combined immune support index: \(2x_0 + 9x_1 \leq 56\)
  11. Upper bound on total combined muscle growth index: \(10x_0 + 3x_1 \leq 55\)

Given these constraints, we can formulate the problem in Gurobi as follows:

```python
from gurobipy import *

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

# Define variables
x0 = m.addVar(vtype=GRB.CONTINUOUS, name="milligrams_of_iron")
x1 = m.addVar(vtype=GRB.INTEGER, name="milligrams_of_potassium")

# Objective function
m.setObjective(5*x0**2 + 4*x0*x1 + 3*x1**2 + 8*x0 + 7*x1, GRB.MINIMIZE)

# Constraints
m.addConstr(2**2 * x0**2 + 9**2 * x1**2 >= 48, name="immune_support_index_squared")
m.addConstr(2*x0 + 9*x1 >= 48, name="total_immune_support_index")
m.addConstr(10*x0 + 3*x1 >= 18, name="muscle_growth_index")
m.addConstr(-6*x0 + 5*x1 >= 0, name="linear_constraint")
m.addConstr(2*x0 + 9*x1 <= 56, name="upper_bound_immune_support")
m.addConstr(10*x0 + 3*x1 <= 55, name="upper_bound_muscle_growth")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of iron: {x0.x}")
    print(f"Milligrams of potassium: {x1.x}")
else:
    print("No optimal solution found.")
```

This code defines the variables, objective function, and constraints as described in the problem statement. It then optimizes the model and prints out the solution if one is found. Note that Gurobi's `GRB.MINIMIZE` constant is used to specify that we want to minimize the objective function, and `GRB.INTEGER` is used for \(x_1\) (milligrams of potassium) since it must be an integer, while `GRB.CONTINUOUS` is used for \(x_0\) (milligrams of iron) as it can be any real number.