To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the problem statement.

Let's denote:
- \(x_1\) as the number of red umbrellas displayed and sold.
- \(x_2\) as the number of blue umbrellas displayed and sold.

The objective is to maximize profit. Given that a profit of $3 is made on each red umbrella and a profit of $5 is made on each blue umbrella, the objective function can be written as:
\[ \text{Maximize:} \quad 3x_1 + 5x_2 \]

The constraints are as follows:
1. The total number of umbrellas displayed and sold cannot exceed 100:
\[ x_1 + x_2 \leq 100 \]
2. A minimum of 10 umbrellas displayed must be red:
\[ x_1 \geq 10 \]
3. At least 4 times as many customers prefer blue umbrellas to red umbrellas, implying the number of blue umbrellas should be at least 4 times the number of red umbrellas:
\[ x_2 \geq 4x_1 \]

All variables \(x_1\) and \(x_2\) must be non-negative since they represent quantities of umbrellas.

Thus, the symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'number of red umbrellas'), ('x2', 'number of blue umbrellas')],
  'objective_function': '3*x1 + 5*x2',
  'constraints': ['x1 + x2 <= 100', 'x1 >= 10', 'x2 >= 4*x1', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this optimization problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

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

# Create variables
x1 = m.addVar(vtype=GRB.INTEGER, name="red_umbrellas", lb=0)
x2 = m.addVar(vtype=GRB.INTEGER, name="blue_umbrellas", lb=0)

# Set the objective function
m.setObjective(3*x1 + 5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 100, "total_umbrellas")
m.addConstr(x1 >= 10, "min_red_umbrellas")
m.addConstr(x2 >= 4*x1, "blue_to_red_ratio")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Red Umbrellas: {x1.x}")
    print(f"Blue Umbrellas: {x2.x}")
    print(f"Total Profit: ${3*x1.x + 5*x2.x}")
else:
    print("No optimal solution found")
```