## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Peggy', 'hours worked by Ringo', and 'hours worked by Paul', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_0$ : hours worked by Peggy
- $x_1$ : hours worked by Ringo
- $x_2$ : hours worked by Paul

## 3: Define the objective function in symbolic notation
The objective function to minimize is $8.67x_0^2 + 2.18x_0x_1 + 8.93x_0x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $17x_0 \leq 236$ (Peggy's computer competence rating constraint, but it seems this should be an equality or directly given as $x_0$ has a competence rating of 17)
- $22x_0 \leq 284$ (Peggy's likelihood to quit index constraint, similarly, it seems this should reflect $x_0$'s index directly)
- $5x_1 \leq 236$ (Ringo's computer competence rating)
- $22x_1 \leq 284$ (Ringo's likelihood to quit index)
- $4x_2 \leq 236$ (Paul's computer competence rating)
- $11x_2 \leq 284$ (Paul's likelihood to quit index)
- $17x_0^2 + 5x_1^2 \geq 33$ (combined computer competence rating from $x_0$ and $x_1$)
- $17x_0^2 + 4x_2^2 \geq 30$ (combined computer competence rating from $x_0$ and $x_2$)
- $17x_0 + 5x_1 + 4x_2 \geq 69$ (total combined computer competence rating)
- $22x_0 + 22x_1 + 11x_2 \geq 86$ (total combined likelihood to quit index)
- $22x_0^2 + 22x_1^2 \geq 86$ (combined likelihood to quit index from $x_0$ and $x_1$)
- $22x_0 + 22x_1 + 11x_2 \geq 86$ (already listed)
- $x_0 - 4x_1 \geq 0$ (relationship between $x_0$ and $x_1$)
- $-5x_0^2 + 9x_2^2 \geq 0$ (relationship between $x_0$ and $x_2$)
- $17x_0 + 5x_1 + 4x_2 \leq 114$ (upper limit on total computer competence)
- $22x_0 + 22x_1 \leq 238$ (limit on combined likelihood to quit index from $x_0$ and $x_1$)
- $22x_1 + 11x_2 \leq 132$ (limit on combined likelihood to quit index from $x_1$ and $x_2$)
- $22^2x_0^2 + 22^2x_1^2 + 11^2x_2^2 \leq 140$ (limit on squared terms of likelihood to quit index)

## 5: Correct and Simplify Constraints
Correcting and simplifying based on the problem statement:
- Direct constraints: $x_0 = 17/17 = 1$ type equality for direct assignments are not properly represented. Let's focus on given constraints directly from the problem:
  - The total combined computer competence rating from hours worked by Peggy squared plus hours worked by Ringo squared: $17^2x_0^2 + 5^2x_1^2 \geq 33$
  - The total combined computer competence rating from hours worked by Peggy squared, and hours worked by Paul squared: $17^2x_0^2 + 4^2x_2^2 \geq 30$
  - The total combined computer competence rating from hours worked by Peggy plus hours worked by Ringo plus hours worked by Paul: $17x_0 + 5x_1 + 4x_2 \geq 69$
  - And similarly for likelihood to quit.

## 6: Formulate the Problem in Gurobi
Given the complexity and potential for error in manual transcription, let's focus on the Gurobi formulation directly.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_Peggy", lb=0)  # hours worked by Peggy
x1 = m.addVar(name="hours_worked_by_Ringo", lb=0)  # hours worked by Ringo
x2 = m.addVar(name="hours_worked_by_Paul", lb=0)   # hours worked by Paul

# Objective function
m.setObjective(8.67 * x0**2 + 2.18 * x0 * x1 + 8.93 * x0 * x2, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(17 * x0 <= 236)  # Peggy computer competence
m.addConstr(22 * x0 <= 284)  # Peggy likelihood to quit
m.addConstr(5 * x1 <= 236)   # Ringo computer competence
m.addConstr(22 * x1 <= 284)  # Ringo likelihood to quit
m.addConstr(4 * x2 <= 236)   # Paul computer competence
m.addConstr(11 * x2 <= 284)  # Paul likelihood to quit

m.addConstr(17**2 * x0**2 + 5**2 * x1**2 >= 33)  # combined computer competence from x0, x1
m.addConstr(17**2 * x0**2 + 4**2 * x2**2 >= 30)  # combined computer competence from x0, x2
m.addConstr(17 * x0 + 5 * x1 + 4 * x2 >= 69)      # total combined computer competence

m.addConstr(22 * x0 + 22 * x1 + 11 * x2 >= 86)   # total combined likelihood to quit
m.addConstr(22**2 * x0**2 + 22**2 * x1**2 >= 86) # combined likelihood to quit from x0, x1

m.addConstr(x0 - 4 * x1 >= 0)                  # relationship between x0 and x1
m.addConstr(-5 * x0**2 + 9 * x2**2 >= 0)       # relationship between x0 and x2

m.addConstr(17 * x0 + 5 * x1 + 4 * x2 <= 114)   # upper limit on total computer competence
m.addConstr(22 * x0 + 22 * x1 <= 238)           # limit on combined likelihood to quit from x0, x1
m.addConstr(22 * x1 + 11 * x2 <= 132)           # limit on combined likelihood to quit from x1, x2

# Solve the model
m.optimize()

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

## 7: Symbolic Representation
The symbolic representation is as follows:

```json
{
    'sym_variables': [('x0', 'hours worked by Peggy'), ('x1', 'hours worked by Ringo'), ('x2', 'hours worked by Paul')],
    'objective_function': '8.67*x0^2 + 2.18*x0*x1 + 8.93*x0*x2',
    'constraints': [
        '17*x0 <= 236',
        '22*x0 <= 284',
        '5*x1 <= 236',
        '22*x1 <= 284',
        '4*x2 <= 236',
        '11*x2 <= 284',
        '289*x0^2 + 25*x1^2 >= 33',
        '289*x0^2 + 16*x2^2 >= 30',
        '17*x0 + 5*x1 + 4*x2 >= 69',
        '22*x0 + 22*x1 + 11*x2 >= 86',
        '484*x0^2 + 484*x1^2 >= 86',
        'x0 - 4*x1 >= 0',
        '-5*x0^2 + 9*x2^2 >= 0',
        '17*x0 + 5*x1 + 4*x2 <= 114',
        '22*x0 + 22*x1 <= 238',
        '22*x1 + 11*x2 <= 132',
        '484*x0^2 + 484*x1^2 + 121*x2^2 <= 140'
    ]
}
```