## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Peggy', 'hours worked by Jean', and 'hours worked by Dale'. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $4x_0^2 + 9x_1x_2 + 6x_1$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $x_0 \geq 0$, $x_1 \geq 0$, $x_2 \geq 0$ ( implicit non-negativity constraints)
- $x_0 = 18 / 18 = 1$ is not a constraint but a fixed value, similarly $x_1 = 23 / 23 = 1$ and $x_2 = 7 / 7 = 1$ are not constraints but given values.
- $x_0^2 + x_2^2 \leq 209$
- $x_0 + x_1 \leq 134 / 18 = 7.444$ is incorrect, it should be $x_0 + x_1 \leq 134$ is not directly given, the correct interpretation is $18x_0 + 23x_1 \leq 254$ and $7x_2 \leq 254$ but there are specific constraints for organization score and dollar cost.
- Correct interpretation of constraints:
  - Organization score constraints:
    - $18x_0 \leq 254$
    - $23x_1 \leq 253$
    - $7x_2 \leq 254$
    - $x_0^2 + x_2^2 \leq 209$
    - $18x_0 + 23x_1 \leq 134$ 
    - $18x_0 + 23x_1 + 7x_2 \leq 134$
  - Dollar cost per hour constraints:
    - $11x_0 \leq 253$
    - $23x_1 \leq 253$
    - $26x_2 \leq 253$
    - $23x_1 + 26x_2 \leq 210$
    - $11x_0 + 26x_2 \leq 198$
    - $11x_0 + 23x_1 + 26x_2 \leq 198$

## 4: Convert the problem into a Gurobi code
We will use Gurobi to solve this optimization problem.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_Peggy", lb=0)
x1 = m.addVar(name="hours_worked_by_Jean", lb=0)
x2 = m.addVar(name="hours_worked_by_Dale", lb=0)

# Objective function
m.setObjective(4 * x0**2 + 9 * x1 * x2 + 6 * x1, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(18 * x0 <= 254)
m.addConstr(23 * x1 <= 253)
m.addConstr(7 * x2 <= 254)
m.addConstr(x0**2 + x2**2 <= 209)
m.addConstr(18 * x0 + 23 * x1 <= 134)
m.addConstr(18 * x0 + 23 * x1 + 7 * x2 <= 134)
m.addConstr(23 * x1 + 26 * x2 <= 210)
m.addConstr(11 * x0 + 26 * x2 <= 198)
m.addConstr(11 * x0 + 23 * x1 + 26 * x2 <= 198)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Peggy: {x0.varValue}")
    print(f"Hours worked by Jean: {x1.varValue}")
    print(f"Hours worked by Dale: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'hours worked by Peggy'), ('x1', 'hours worked by Jean'), ('x2', 'hours worked by Dale')],
    'objective_function': '4*x0^2 + 9*x1*x2 + 6*x1',
    'constraints': [
        '18*x0 <= 254',
        '23*x1 <= 253',
        '7*x2 <= 254',
        'x0^2 + x2^2 <= 209',
        '18*x0 + 23*x1 <= 134',
        '18*x0 + 23*x1 + 7*x2 <= 134',
        '23*x1 + 26*x2 <= 210',
        '11*x0 + 26*x2 <= 198',
        '11*x0 + 23*x1 + 26*x2 <= 198'
    ]
}
```