## Step 1: Define the variables and the objective function
The variables are 'hours worked by John', 'hours worked by Paul', 'hours worked by Bobby', and 'hours worked by Dale'. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively. The objective function to maximize is $4x_0^2 + 7x_0x_1 + 4x_0x_2 + 3x_0x_3 + 6x_1x_3 + x_2x_3 + 5x_3^2$.

## Step 2: Define the constraints
The constraints given are:
- $x_0$ has an organization score of 8.
- $x_0$ has a computer competence rating of 3.
- $x_1$ has an organization score of 1.
- $x_1$ has a computer competence rating of 4.
- $x_2$ has an organization score of 6.
- $x_2$ has a computer competence rating of 3.
- $x_3$ has an organization score of 8.
- $x_3$ has a computer competence rating of 5.
- $3x_0 + 3x_2 \geq 8$.
- $3x_0 + 4x_1 + 3x_2 \geq 12$.
- $-10x_0 + 9x_2 \geq 0$.
- $x_1^2 + x_3^2 \leq 76$.
- $x_2^2 + x_3^2 \leq 64$.
- $8x_0^2 + 6x_2^2 \leq 103$.
- $8x_0^2 + 8x_3^2 \leq 121$.
- $x_1 + 6x_2 \leq 88$.
- $8x_0 + x_1 + 6x_2 + 8x_3 \leq 88$.
- $3x_0 + 4x_1 \leq 79$.
- $3x_2 + 5x_3 \leq 39$.
- $9x_0^2 + 9x_2^2 \leq 27$.
- $9x_0^2 + 25x_3^2 \leq 27$.
- $9x_0^2 + 9x_2^2 + 25x_3^2 \leq 72$.
- $3x_0 + 4x_1 + 5x_3 \leq 20$.
- $16x_1^2 + 9x_2^2 + 25x_3^2 \leq 73$.
- $3x_0 + 4x_1 + 3x_2 + 5x_3 \leq 73$.

## 3: Implement the problem in Gurobi
We will use Gurobi's Python interface to model and solve this problem.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_John", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x1 = m.addVar(name="hours_worked_by_Paul", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x2 = m.addVar(name="hours_worked_by_Bobby", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x3 = m.addVar(name="hours_worked_by_Dale", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)

# Objective function
m.setObjective(4*x0**2 + 7*x0*x1 + 4*x0*x2 + 3*x0*x3 + 6*x1*x3 + x2*x3 + 5*x3**2, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(x0 == 8, name="John_organization_score")
m.addConstr(3*x0 == 3, name="John_computer_competence_rating")
m.addConstr(x1 == 1, name="Paul_organization_score")
m.addConstr(4*x1 == 4, name="Paul_computer_competence_rating")
m.addConstr(6*x2 == 6, name="Bobby_organization_score")
m.addConstr(3*x2 == 3, name="Bobby_computer_competence_rating")
m.addConstr(8*x3 == 8, name="Dale_organization_score")
m.addConstr(5*x3 == 5, name="Dale_computer_competence_rating")

m.addConstr(3*x0 + 3*x2 >= 8, name="combined_computer_rating_1")
m.addConstr(3*x0 + 4*x1 + 3*x2 >= 12, name="combined_computer_rating_2")
m.addConstr(-10*x0 + 9*x2 >= 0, name="john_bobby_hours")

m.addConstr(x1**2 + x3**2 <= 76, name="paul_dale_organization")
m.addConstr(x2**2 + x3**2 <= 64, name="bobby_dale_organization")
m.addConstr(8*x0**2 + 6*x2**2 <= 103, name="john_bobby_organization")
m.addConstr(8*x0**2 + 8*x3**2 <= 121, name="john_dale_organization")
m.addConstr(x1 + 6*x2 <= 88, name="paul_bobby_organization")
m.addConstr(8*x0 + x1 + 6*x2 + 8*x3 <= 88, name="total_organization")

m.addConstr(3*x0 + 4*x1 <= 79, name="john_paul_computer")
m.addConstr(3*x2 + 5*x3 <= 39, name="bobby_dale_computer")

m.addConstr(9*x0**2 + 9*x2**2 <= 27, name="john_bobby_computer_squared")
m.addConstr(9*x0**2 + 25*x3**2 <= 27, name="john_dale_computer_squared")
m.addConstr(9*x0**2 + 9*x2**2 + 25*x3**2 <= 72, name="total_computer_squared")

m.addConstr(3*x0 + 4*x1 + 5*x3 <= 20, name="john_paul_dale_computer")

m.addConstr(16*x1**2 + 9*x2**2 + 25*x3**2 <= 73, name="paul_bobby_dale_computer_squared")
m.addConstr(3*x0 + 4*x1 + 3*x2 + 5*x3 <= 73, name="total_computer")

# Correct the initial constraints for organization and computer competence ratings
m.addConstr(x0 == 8, name="john_org")
m.addConstr(3*x0 == 3, name="john_comp")
m.addConstr(x1 == 1, name="paul_org")
m.addConstr(4*x1 == 4, name="paul_comp")
m.addConstr(6*x2 == 6, name="bobby_org")
m.addConstr(3*x2 == 3, name="bobby_comp")
m.addConstr(8*x3 == 8, name="dale_org")
m.addConstr(5*x3 == 5, name="dale_comp")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by John: ", x0.varValue)
    print("Hours worked by Paul: ", x1.varValue)
    print("Hours worked by Bobby: ", x2.varValue)
    print("Hours worked by Dale: ", x3.varValue)
else:
    print("The model is infeasible")
```