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

Let's denote the number of dining tables as \(x_1\) and the number of bed frames as \(x_2\). The objective is to maximize profit, where the profit per dining table is $300 and per bed frame is $400. The constraints are based on the availability of oak and mahogany wood.

## Step 2: Formulate the objective function

The objective function to maximize profit is: \(300x_1 + 400x_2\).

## Step 3: Formulate the constraints

Each dining table requires 3 units of oak wood and each bed frame requires 5 units of oak wood, with 300 units available. This gives the constraint: \(3x_1 + 5x_2 \leq 300\).

Each dining table requires 5 units of mahogany wood and each bed frame requires 2 units of mahogany wood, with 400 units available. This gives the constraint: \(5x_1 + 2x_2 \leq 400\).

Also, \(x_1 \geq 0\) and \(x_2 \geq 0\), since the number of tables and frames cannot be negative.

## 4: Symbolic representation in the required format

```json
{
    'sym_variables': [('x1', 'dining tables'), ('x2', 'bed frames')],
    'objective_function': '300*x1 + 400*x2',
    'constraints': [
        '3*x1 + 5*x2 <= 300',
        '5*x1 + 2*x2 <= 400',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Step 5: Convert the problem into Gurobi code

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="dining_tables", lb=0, vtype=gurobi.GRB.CONTINUOUS)
    x2 = model.addVar(name="bed_frames", lb=0, vtype=gurobi.GRB.CONTINUOUS)

    # Set the objective function
    model.setObjective(300*x1 + 400*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(3*x1 + 5*x2 <= 300, name="oak_wood_constraint")
    model.addConstr(5*x1 + 2*x2 <= 400, name="mahogany_wood_constraint")

    # Optimize the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of dining tables: {x1.varValue}")
        print(f"Number of bed frames: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```