To solve this optimization problem, we first need to understand the objective function and the constraints provided. The objective function to maximize is:

\[ 6H^2 + 6H \cdot P + 9H \cdot M + 2P^2 + 4P \cdot M + 9P \cdot D + 5D^2 + 7H + 7P + 7M \]

where \(H\), \(P\), \(M\), and \(D\) represent the hours worked by Hank, Peggy, Mary, and Dale, respectively.

## Step 1: Define the Variables
We start by defining the variables for the hours worked by each person.

## Step 2: Implement the Objective Function
The objective function needs to be maximized. In Gurobi, this can be achieved by setting the objective function using the `setObjective` method.

## 3: Define the Constraints
There are numerous constraints provided, including constraints on productivity ratings, likelihood to quit indices, dollar costs per hour, computer competence ratings, and work quality ratings. Each of these constraints needs to be translated into a mathematical expression and then implemented in Gurobi.

## 4: Gurobi Code Implementation
Given the complexity and the number of constraints, we will directly implement the problem in Gurobi Python.

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
H = m.addVar(name="H", lb=0)  # Hours worked by Hank
P = m.addVar(name="P", lb=0)  # Hours worked by Peggy
M = m.addVar(name="M", lb=0)  # Hours worked by Mary
D = m.addVar(name="D", lb=0, integrality=gp.GRB.INTEGER)  # Hours worked by Dale

# Objective function
m.setObjective(6*H**2 + 6*H*P + 9*H*M + 2*P**2 + 4*P*M + 9*P*D + 5*D**2 + 7*H + 7*P + 7*M, gp.GRB.MAXIMIZE)

# Constraints
# Productivity rating constraints
m.addConstr(7.78*H <= 198)
m.addConstr(4.57*P <= 198)
m.addConstr(2.91*M <= 198)
m.addConstr(5.28*D <= 198)

# Likelihood to quit index constraints
m.addConstr(7.7*H <= 197)
m.addConstr(8.32*P <= 197)
m.addConstr(6.3*M <= 197)
m.addConstr(8.37*D <= 197)

# Dollar cost per hour constraints
m.addConstr(1.01*H <= 122)
m.addConstr(0.91*P <= 122)
m.addConstr(8.59*M <= 122)
m.addConstr(4.06*D <= 122)

# Computer competence rating constraints
m.addConstr(6.02*H <= 182)
m.addConstr(6.71*P <= 182)
m.addConstr(2.0*M <= 182)
m.addConstr(6.5*D <= 182)

# Work quality rating constraints
m.addConstr(3.38*H <= 123)
m.addConstr(8.21*P <= 123)
m.addConstr(8.49*M <= 123)
m.addConstr(6.87*D <= 123)

# Additional constraints
m.addConstr((2.91*M)**2 + (5.28*D)**2 >= 18)
m.addConstr(7.78*H + 5.28*D >= 49)
m.addConstr((7.78*H)**2 + (2.91*M)**2 + (5.28*D)**2 >= 31)
m.addConstr(7.78*H + 4.57*P + 5.28*D >= 31)
m.addConstr(7.78*H + 2.91*M + 5.28*D >= 27)

# Likelihood to quit index constraints continued
m.addConstr((7.7*H)**2 + (8.32*P)**2 >= 43)
m.addConstr(8.32*P + 8.37*D >= 17)
m.addConstr(6.3*M + 8.37*D >= 29)
m.addConstr(7.7*H + 6.3*M >= 32)
m.addConstr((8.32*P)**2 + (6.3*M)**2 >= 25)
m.addConstr(8.32*P*M + 8.37*D*M >= 41)

# Dollar cost per hour constraints continued
m.addConstr(1.01*H + 8.59*M >= 24)
m.addConstr((0.91*P)**2 + (4.06*D)**2 >= 13)
m.addConstr(1.01*H + 4.06*D >= 19)
m.addConstr(8.59*M + 4.06*D >= 18)

# Computer competence rating constraints continued
m.addConstr((6.71*P)**2 + (2.0*M)**2 >= 16)
m.addConstr((6.02*H)**2 + (6.5*D)**2 >= 29)
m.addConstr((2.0*M)**2 + (6.5*D)**2 >= 42)
m.addConstr(6.02*H + 2.0*M >= 32)

# Work quality rating constraints continued
m.addConstr((8.21*P)**2 + (8.49*M)**2 + (6.87*D)**2 >= 16)
m.addConstr((3.38*H)**2 + (8.21*P)**2 + (8.49*M)**2 >= 16)

# Upper bound constraints
m.addConstr(4.57*P + 5.28*D <= 113)
m.addConstr(2.91*M + 5.28*D <= 109)
m.addConstr((7.78*H)**2 + (2.91*M)**2 <= 174)
m.addConstr(4.57*P*2.91*M <= 117)
m.addConstr(7.78*H + 5.28*D <= 139)
m.addConstr((4.57*P)**2 + (2.91*M)**2 + (5.28*D)**2 <= 132)
m.addConstr(7.78*H + 4.57*P + 5.28*D <= 195)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Hours worked by Hank:", H.varValue)
    print("Hours worked by Peggy:", P.varValue)
    print("Hours worked by Mary:", M.varValue)
    print("Hours worked by Dale:", D.varValue)
else:
    print("The model is infeasible")
```