## Step 1: Define the variables and the objective function
The variables are 'hours worked by Dale', 'hours worked by Peggy', 'hours worked by Paul', and 'hours worked by John'. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively. The objective function to minimize is:
\[ \text{minimize:} \quad 1 \cdot x_0 \cdot x_1 + 1 \cdot x_0 \cdot x_2 + 3 \cdot x_0 \cdot x_3 + 4 \cdot x_1^2 + 8 \cdot x_1 \cdot x_2 + 6 \cdot x_1 \cdot x_3 + 7 \cdot x_2 \cdot x_3 + 1 \cdot x_3^2 + 4 \cdot x_0 + 8 \cdot x_1 + 2 \cdot x_2 + 2 \cdot x_3 \]

## Step 2: Define the constraints
The constraints given are:
- $x_0$ is an integer, $x_1$ is an integer, $x_2$ is an integer, $x_3$ is an integer.
- Computer competence ratings: $x_0 \cdot 5$, $x_1 \cdot 8$, $x_2 \cdot 11$, $x_3 \cdot 4$.
- Paperwork competence ratings: $x_0 \cdot 7$, $x_1 \cdot 2$, $x_2 \cdot 2$, $x_3 \cdot 9$.
- Specific constraints:
1. $11x_2 + 4x_3 \geq 25$
2. $25x_0^2 + 121x_2^2 \geq 17$
3. $8x_1 + 4x_3 \geq 25$
4. $5x_0 + 4x_3 \geq 10$
5. $5x_0 + 8x_1 + 11x_2 \geq 24$
6. $5x_0 + 8x_1 + 11x_2 + 4x_3 \geq 24$
7. $2x_1 + 9x_3 \geq 31$
8. $7x_0 + 9x_3 \geq 16$
9. $7x_0 + 2x_1 + 2x_2 \geq 31$
10. $7x_0 + 2x_1 + 2x_2 + 9x_3 \geq 31$
11. $2x_1 + 2x_2 \leq 80$
12. $7x_0 + 9x_3 \leq 67$
13. $49x_0^2 + 4x_2^2 + 81x_3^2 \leq 55$
14. $7x_0 + 2x_1 + 2x_2 \leq 53$
15. $2x_1 + 2x_2 + 9x_3 \leq 88$

## 3: Implement the problem in Gurobi
To solve this problem, we will use the Gurobi Python API.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_Dale", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="hours_worked_by_Peggy", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="hours_worked_by_Paul", vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="hours_worked_by_John", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(1 * x0 * x1 + 1 * x0 * x2 + 3 * x0 * x3 + 4 * x1**2 + 8 * x1 * x2 + 6 * x1 * x3 + 7 * x2 * x3 + 1 * x3**2 + 4 * x0 + 8 * x1 + 2 * x2 + 2 * x3, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(11 * x2 + 4 * x3 >= 25)
m.addConstr(25 * x0**2 + 121 * x2**2 >= 17)
m.addConstr(8 * x1 + 4 * x3 >= 25)
m.addConstr(5 * x0 + 4 * x3 >= 10)
m.addConstr(5 * x0 + 8 * x1 + 11 * x2 >= 24)
m.addConstr(5 * x0 + 8 * x1 + 11 * x2 + 4 * x3 >= 24)
m.addConstr(2 * x1 + 9 * x3 >= 31)
m.addConstr(7 * x0 + 9 * x3 >= 16)
m.addConstr(7 * x0 + 2 * x1 + 2 * x2 >= 31)
m.addConstr(7 * x0 + 2 * x1 + 2 * x2 + 9 * x3 >= 31)
m.addConstr(2 * x1 + 2 * x2 <= 80)
m.addConstr(7 * x0 + 9 * x3 <= 67)
m.addConstr(49 * x0**2 + 4 * x2**2 + 81 * x3**2 <= 55)
m.addConstr(7 * x0 + 2 * x1 + 2 * x2 <= 53)
m.addConstr(2 * x1 + 2 * x2 + 9 * x3 <= 88)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Dale: {x0.varValue}")
    print(f"Hours worked by Peggy: {x1.varValue}")
    print(f"Hours worked by Paul: {x2.varValue}")
    print(f"Hours worked by John: {x3.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```