## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Laura' and 'hours worked by Ringo', which we can denote as $x_0$ and $x_1$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $3x_0^2 + 2x_0x_1 + 4x_1^2 + 7x_0 + 8x_1$.

## Step 3: List the constraints
The constraints are:
- $15x_0 = 15$ (Laura's computer competence rating is 15, but this seems to be an equality constraint without hours, so it might be a fixed value or misinterpreted)
- $18x_0 = 18$ (Laura's likelihood to quit index is 18, similar issue)
- $7x_0 = 7$ (Laura's organization score is 7)
- $7x_0 = 7$ (Laura's paperwork competence rating is 7)
- $28x_0 = 28$ (Laura's productivity rating is 28)
- $6x_1 = 6$ (Ringo's computer competence rating is 6)
- $20x_1 = 20$ (Ringo's likelihood to quit index is 20)
- $3x_1 = 3$ (Ringo's organization score is 3)
- $25x_1 = 25$ (Ringo's paperwork competence rating is 25)
- $10x_1 = 10$ (Ringo's productivity rating is 10)
- $15x_0 + 6x_1 \geq 32$ (total combined computer competence rating)
- $18x_0 + 20x_1 \geq 31$ (total combined likelihood to quit index)
- $7x_0^2 + 3x_1^2 \geq 53$ (total combined organization score)
- $7x_0^2 + 25x_1^2 \geq 12$ (total combined paperwork competence rating)
- $28x_0 + 10x_1 \geq 22$ (total combined productivity rating)
- $-7x_0 + 3x_1 \geq 0$
- $15x_0 + 6x_1 \leq 111$
- $18x_0 + 20x_1 \leq 60$
- $7x_0^2 + 3x_1^2 \leq 59$
- $7x_0^2 + 3x_1^2 \leq 59$ (duplicate, but keeping for completeness)
- $7x_0 + 25x_1 \leq 43$ (corrected to reflect paperwork)
- $28x_0^2 + 10x_1^2 \leq 63$
- $28x_0 + 10x_1 \leq 63$

However, upon closer inspection, the initial equality constraints seem to be misstated as they do not involve $x_0$ and $x_1$ in a meaningful way for an optimization problem (they seem to be statements of fact rather than constraints). The actual constraints from the problem statement that involve $x_0$ and $x_1$ directly are:
- $15x_0 + 6x_1 \geq 32$
- $18x_0 + 20x_1 \geq 31$
- $7x_0^2 + 3x_1^2 \geq 53$
- $7x_0^2 + 25x_1^2 \geq 12$
- $28x_0 + 10x_1 \geq 22$
- $-7x_0 + 3x_1 \geq 0$
- $15x_0 + 6x_1 \leq 111$
- $18x_0 + 20x_1 \leq 60$
- $7x_0^2 + 3x_1^2 \leq 59$
- $7x_0 + 25x_1 \leq 43$
- $28x_0^2 + 10x_1^2 \leq 63$
- $28x_0 + 10x_1 \leq 63$

## 4: Correct Interpretation of Initial Conditions
The initial conditions given for Laura and Ringo (e.g., $15, 18, 7, 7, 28$ for Laura and $6, 20, 3, 25, 10$ for Ringo) seem to be descriptive and not directly part of the optimization problem as constraints but rather as attributes.

## 5: Symbolic Representation
The symbolic representation of the problem is:
- Variables: $x_0$ ('hours worked by Laura'), $x_1$ ('hours worked by Ringo')
- Objective Function: $3x_0^2 + 2x_0x_1 + 4x_1^2 + 7x_0 + 8x_1$
- Constraints:
  - $15x_0 + 6x_1 \geq 32$
  - $18x_0 + 20x_1 \geq 31$
  - $7x_0^2 + 3x_1^2 \geq 53$
  - $7x_0^2 + 25x_1^2 \geq 12$
  - $28x_0 + 10x_1 \geq 22$
  - $-7x_0 + 3x_1 \geq 0$
  - $15x_0 + 6x_1 \leq 111$
  - $18x_0 + 20x_1 \leq 60$
  - $7x_0^2 + 3x_1^2 \leq 59$
  - $7x_0 + 25x_1 \leq 43$
  - $28x_0^2 + 10x_1^2 \leq 63$
  - $28x_0 + 10x_1 \leq 63$

## 6: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Laura
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Ringo

    # Objective function
    model.setObjective(3*x0**2 + 2*x0*x1 + 4*x1**2 + 7*x0 + 8*x1, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(15*x0 + 6*x1 >= 32)
    model.addConstr(18*x0 + 20*x1 >= 31)
    model.addConstr(7*x0**2 + 3*x1**2 >= 53)
    model.addConstr(7*x0**2 + 25*x1**2 >= 12)
    model.addConstr(28*x0 + 10*x1 >= 22)
    model.addConstr(-7*x0 + 3*x1 >= 0)
    model.addConstr(15*x0 + 6*x1 <= 111)
    model.addConstr(18*x0 + 20*x1 <= 60)
    model.addConstr(7*x0**2 + 3*x1**2 <= 59)
    model.addConstr(7*x0 + 25*x1 <= 43)
    model.addConstr(28*x0**2 + 10*x1**2 <= 63)
    model.addConstr(28*x0 + 10*x1 <= 63)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("x0: ", x0.varValue)
        print("x1: ", x1.varValue)
    else:
        print("No solution found")

solve_optimization_problem()
```

## 7: Final Representation
```json
{
  "sym_variables": [
    ["x0", "hours worked by Laura"],
    ["x1", "hours worked by Ringo"]
  ],
  "objective_function": "3*x0^2 + 2*x0*x1 + 4*x1^2 + 7*x0 + 8*x1",
  "constraints": [
    "15*x0 + 6*x1 >= 32",
    "18*x0 + 20*x1 >= 31",
    "7*x0^2 + 3*x1^2 >= 53",
    "7*x0^2 + 25*x1^2 >= 12",
    "28*x0 + 10*x1 >= 22",
    "-7*x0 + 3*x1 >= 0",
    "15*x0 + 6*x1 <= 111",
    "18*x0 + 20*x1 <= 60",
    "7*x0^2 + 3*x1^2 <= 59",
    "7*x0 + 25*x1 <= 43",
    "28*x0^2 + 10*x1^2 <= 63",
    "28*x0 + 10*x1 <= 63"
  ]
}
```