## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

## Step 2: Identify the variables and their corresponding symbolic notation
The variables are:
- 'hours worked by Peggy' corresponding to $x_1$
- 'hours worked by Bobby' corresponding to $x_2$
- 'hours worked by Dale' corresponding to $x_3$

## 3: Formulate the objective function using symbolic variables
The objective function to minimize is: $9.64x_1^2 + 7.83x_1x_3 + 4.68x_1 + 2.93x_2 + 6.99x_3$

## 4: List the constraints using symbolic variables
The constraints are:
1. $14x_1 \leq 59$ (Peggy's likelihood to quit index is 14, but this seems to be an attribute rather than a constraint. Assuming it's about the index value itself: $x_1 = 14$ is not a constraint but a given value)
2. $2x_2 \leq 59$ (Bobby's likelihood to quit index is 2)
3. $5x_3 \leq 59$ (Dale's likelihood to quit index is 5)
4. $14x_1^2 + 2x_2^2 \geq 13$
5. $2x_2 + 5x_3 \geq 9$
6. $14x_1 + 2x_2 + 5x_3 \geq 16$
7. $14x_1 + 2x_2 + 5x_3 \geq 16$ (Redundant with constraint 6)
8. $-10x_2^2 + 6x_3^2 \geq 0$
9. $2x_2 + 5x_3 \leq 20$
10. $14x_1 + 2x_2 \leq 53$
11. $14x_1 + 2x_2 + 5x_3 \leq 43$
12. $x_1$ is an integer
13. $x_2$ is an integer

## 5: Correct and clarify constraints based on problem description
Given attributes:
- Peggy's likelihood to quit index: $14$
- Bobby's likelihood to quit index: $2$
- Dale's likelihood to quit index: $5$

These seem to directly provide values rather than constraints. The actual constraints from the problem statement need to be directly translated:
- The total combined likelihood to quit index from hours worked by Peggy squared, and hours worked by Bobby squared has to be at least 13: $14x_1^2 + 2x_2^2 \geq 13$
- ... (rest as listed)

## 6: Symbolic representation
```json
{
    'sym_variables': [('x1', 'hours worked by Peggy'), ('x2', 'hours worked by Bobby'), ('x3', 'hours worked by Dale')],
    'objective_function': '9.64*x1^2 + 7.83*x1*x3 + 4.68*x1 + 2.93*x2 + 6.99*x3',
    'constraints': [
        '14*x1^2 + 2*x2^2 >= 13',
        '2*x2 + 5*x3 >= 9',
        '14*x1 + 2*x2 + 5*x3 >= 16',
        '-10*x2^2 + 6*x3^2 >= 0',
        '2*x2 + 5*x3 <= 20',
        '14*x1 + 2*x2 <= 53',
        '14*x1 + 2*x2 + 5*x3 <= 43',
        'x1 % 1 == 0',  # x1 is an integer
        'x2 % 1 == 0'   # x2 is an integer
    ]
}
```

## 7: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # hours worked by Bobby
    x3 = model.addVar(name="x3")  # hours worked by Dale

    # Objective function
    model.setObjective(9.64 * x1**2 + 7.83 * x1 * x3 + 4.68 * x1 + 2.93 * x2 + 6.99 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(14 * x1**2 + 2 * x2**2 >= 13)
    model.addConstr(2 * x2 + 5 * x3 >= 9)
    model.addConstr(14 * x1 + 2 * x2 + 5 * x3 >= 16)
    model.addConstr(-10 * x2**2 + 6 * x3**2 >= 0)
    model.addConstr(2 * x2 + 5 * x3 <= 20)
    model.addConstr(14 * x1 + 2 * x2 <= 53)
    model.addConstr(14 * x1 + 2 * x2 + 5 * x3 <= 43)

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Peggy: {x1.varValue}")
        print(f"Hours worked by Bobby: {x2.varValue}")
        print(f"Hours worked by Dale: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

solve_optimization_problem()
```