## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the amount of cream Alpha in mg.
- $x_2$ represents the amount of cream Beta in mg.

## Step 2: Formulate the objective function
The objective is to minimize the cost of the mixture. The cost of cream Alpha is $0.70 per mg, and the cost of cream Beta is $0.90 per mg. Therefore, the objective function can be represented as:
\[ \text{Minimize:} \quad 0.70x_1 + 0.90x_2 \]

## 3: Define the constraints
The mixture must contain at least 4 units of compound X and 8 units of compound Y.
- The amount of compound X in cream Alpha is 2 units/mg, and in cream Beta is 4.1 units/mg. So, the constraint for compound X is:
\[ 2x_1 + 4.1x_2 \geq 4 \]
- The amount of compound Y in cream Alpha is 2.7 units/mg, and in cream Beta is 3.2 units/mg. So, the constraint for compound Y is:
\[ 2.7x_1 + 3.2x_2 \geq 8 \]
Additionally, the amounts of cream Alpha and Beta cannot be negative:
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

## 4: Symbolic representation of the problem
The symbolic representation is:
```json
{
'sym_variables': [('x1', 'cream Alpha'), ('x2', 'cream Beta')],
'objective_function': '0.70*x1 + 0.90*x2',
'constraints': [
    '2*x1 + 4.1*x2 >= 4',
    '2.7*x1 + 3.2*x2 >= 8',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 5: Gurobi code
Now, let's implement this problem using Gurobi in Python:
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="x1", lb=0, ub=gurobi.GRB.INFINITY)
    x2 = model.addVar(name="x2", lb=0, ub=gurobi.GRB.INFINITY)

    # Define the objective function
    model.setObjective(0.70 * x1 + 0.90 * x2, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(2 * x1 + 4.1 * x2 >= 4, name="compound_X")
    model.addConstr(2.7 * x1 + 3.2 * x2 >= 8, name="compound_Y")

    # Optimize the model
    model.optimize()

    # Print the status
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Amount of cream Alpha: {x1.varValue} mg")
        print(f"Amount of cream Beta: {x2.varValue} mg")
        print(f"Minimum cost: ${model.objVal:.2f}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("No feasible solution exists.")
    else:
        print("Optimization failed.")

solve_optimization_problem()
```