## Step 1: Define the symbolic representation of the problem

Let's denote the number of hardwood flooring as $x_1$ and the number of carpet as $x_2$. The objective is to maximize profit, which is $400x_1 + 650x_2$. The constraints are based on the available hours for cutting and installation. For cutting, it takes 1 hour for hardwood flooring and 0.5 hours for carpet, with 200 hours available: $1x_1 + 0.5x_2 \leq 200$. For installation, it takes 3 hours for hardwood flooring and 4 hours for carpet, with 400 hours available: $3x_1 + 4x_2 \leq 400$. Also, $x_1 \geq 0$ and $x_2 \geq 0$ because the number of flooring and carpet cannot be negative.

## 2: Write down the symbolic representation in the required format

```json
{
    'sym_variables': [('x1', 'hardwood flooring'), ('x2', 'carpet')],
    'objective_function': '400*x1 + 650*x2',
    'constraints': [
        '1*x1 + 0.5*x2 <= 200',
        '3*x1 + 4*x2 <= 400',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## 3: Translate the symbolic representation into Gurobi code

To solve this problem using Gurobi in Python, we need to install the Gurobi library and have a Gurobi license. The following Python code represents the optimization problem:

```python
import gurobi as gp

# Create a new model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="hardwood_flooring", lb=0, vtype=gp.GRB.CONTINUOUS)
x2 = m.addVar(name="carpet", lb=0, vtype=gp.GRB.CONTINUOUS)

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

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

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hardwood flooring: {x1.varValue}")
    print(f"Carpet: {x2.varValue}")
    print(f"Max Profit: {m.objVal}")
else:
    print("No optimal solution found.")
```