To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining the variables, the objective function, and the constraints using algebraic terms.

Given:
- Variables: 'hours worked by Dale' (let's denote this as $x_0$), 'hours worked by Paul' (denoted as $x_1$)
- Objective Function: Minimize $6 \cdot x_0 \cdot x_1 + 2 \cdot x_1$
- Constraints:
  1. Productivity rating constraints for Dale and Paul.
  2. Organization score constraints for Dale and Paul.
  3. Combined productivity rating constraint.
  4. Combined organization score constraint (squared terms).
  5. Constraint involving squared terms of hours worked by Dale and Paul.
  6. Maximum combined productivity rating constraint.
  7. Maximum combined organization score constraint.
  8. Integer constraint for hours worked by Dale.

Let's denote $x_0$ as 'hours worked by Dale' and $x_1$ as 'hours worked by Paul'. The symbolic representation of the problem is:

```json
{
  'sym_variables': [('x0', 'hours worked by Dale'), ('x1', 'hours worked by Paul')],
  'objective_function': '6*x0*x1 + 2*x1',
  'constraints': [
    '3*x0 + 17*x1 >= 45', 
    '12*x0 + 6*x1 >= 59', 
    '-4*(x0**2) + 3*(x1**2) >= 0', 
    '3*x0 + 17*x1 <= 79', 
    '12*x0 + 6*x1 <= 146',
    'x0 == int(x0)'
  ]
}
```

To implement this in Gurobi, we need to ensure that the variables are correctly defined and that all constraints are properly formulated. Note that Gurobi supports quadratic constraints directly, which simplifies the implementation of some constraints.

```python
from gurobipy import *

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

# Define variables
x0 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Dale")
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Paul")

# Objective function: Minimize 6*x0*x1 + 2*x1
m.setObjective(6*x0*x1 + 2*x1, GRB.MINIMIZE)

# Constraints
m.addConstr(3*x0 + 17*x1 >= 45, name="combined_productivity_rating_min")
m.addConstr(12*x0 + 6*x1 >= 59, name="combined_organization_score_min")
m.addConstr(-4*(x0**2) + 3*(x1**2) >= 0, name="squared_terms_constraint")
m.addConstr(3*x0 + 17*x1 <= 79, name="combined_productivity_rating_max")
m.addConstr(12*x0 + 6*x1 <= 146, name="combined_organization_score_max")

# Optimize model
m.optimize()

# Print results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print("Obj:", m.objVal)
```