To solve this optimization problem using Gurobi, we first need to define our variables and the objective function, followed by the constraints as specified in the problem description.

We are given two variables: `hours worked by Dale` (let's call it `x0`) and `hours worked by Paul` (let's call it `x1`). The objective is to minimize the function `6*x0*x1 + 2*x1`.

The constraints can be broken down as follows:
- Productivity ratings for Dale and Paul are given, but these seem to be coefficients rather than direct constraints.
- Organization scores for Dale and Paul are also provided as coefficients.
- Total combined productivity rating from hours worked must be at least 45.
- The same condition is repeated; we'll consider it once.
- Total combined organization score from the square of hours worked by Dale and hours worked by Paul should be at least 59.
- Another constraint on total combined organization score, which seems similar but will be addressed in the model directly.
- A quadratic constraint involving hours worked by both individuals must be non-negative.
- The maximum total combined productivity rating is 79.
- The maximum total combined organization score is 146.
- Hours worked by Dale cannot be fractional (integer constraint).
- Hours worked by Paul can be fractional.

Let's translate these into Gurobi code:

```python
from gurobipy import *

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

# Define the variables
x0 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Dale")  # Integer constraint for Dale
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Paul")  # Continuous variable for Paul

# Define the objective function
m.setObjective(6*x0*x1 + 2*x1, GRB.MINIMIZE)

# Constraints
# Total combined productivity rating from hours worked must be at least 45
m.addConstr(3*x0 + 17*x1 >= 45, name="productivity_rating_min")

# Total combined organization score from the square of hours worked by Dale and hours worked by Paul should be at least 59
m.addConstr(12*(x0**2) + 6*(x1**2) >= 59, name="organization_score_squared_min")

# Another constraint on total combined organization score (linear)
m.addConstr(12*x0 + 6*x1 >= 59, name="organization_score_min")

# Quadratic constraint involving hours worked by both individuals must be non-negative
m.addConstr(-4*(x0**2) + 3*(x1**2) >= 0, name="quadratic_constraint")

# Maximum total combined productivity rating is 79
m.addConstr(3*x0 + 17*x1 <= 79, name="productivity_rating_max")

# Maximum total combined organization score is 146
m.addConstr(12*x0 + 6*x1 <= 146, name="organization_score_max")

# Solve the model
m.optimize()

# Print solution if found
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: x0 = {x0.x}, x1 = {x1.x}")
else:
    print("No optimal solution found")
```