To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables for the quantities of hardwood flooring and carpet to be installed, formulating the objective function that represents the total profit, and listing the constraints based on the available hours for cutting and installation.

Let's denote:
- $x_1$ as the number of hardwood floorings to be installed,
- $x_2$ as the number of carpets to be installed.

The profit per hardwood flooring is $400, and the profit per carpet is $650. Therefore, the objective function to maximize the total profit can be written as:
\[ 400x_1 + 650x_2 \]

The constraints are based on the available hours for cutting and installation:
- For cutting: $1x_1 + 0.5x_2 \leq 200$ (since it takes 1 hour to cut one hardwood flooring and 0.5 hours to cut one carpet, with a total of 200 hours available),
- For installation: $3x_1 + 4x_2 \leq 400$ (since it takes 3 hours to install one hardwood flooring and 4 hours to install one carpet, with a total of 400 hours available).

Also, $x_1 \geq 0$ and $x_2 \geq 0$ because the number of installations cannot be negative.

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'hardwood floorings'), ('x2', 'carpets')],
    'objective_function': '400*x1 + 650*x2',
    'constraints': ['x1 + 0.5*x2 <= 200', '3*x1 + 4*x2 <= 400', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we'll use the following code:

```python
from gurobipy import *

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

# Create variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hardwood_floorings", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="carpets", lb=0)

# Set the objective function
m.setObjective(400*x1 + 650*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + 0.5*x2 <= 200, "cutting_hours")
m.addConstr(3*x1 + 4*x2 <= 400, "installation_hours")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hardwood floorings: {x1.x}")
    print(f"Carpets: {x2.x}")
    print(f"Total profit: ${400*x1.x + 650*x2.x:.2f}")
else:
    print("No optimal solution found")
```