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

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is: $1 \cdot x_0^2 + 8 \cdot x_0 \cdot x_1 + 2 \cdot x_0 \cdot x_2 + 9 \cdot x_1^2 + 5 \cdot x_1 \cdot x_2 + 2 \cdot x_2^2 + 9 \cdot x_0 + 2 \cdot x_1$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $17x_0 = r_0$ (work quality rating for Paul)
- $9x_0 = r_1$ (paperwork competence rating for Paul)
- $15x_1 = r_0$ (work quality rating for Hank)
- $9x_1 = r_1$ (paperwork competence rating for Hank)
- $12x_2 = r_0$ (work quality rating for Laura)
- $5x_2 = r_1$ (paperwork competence rating for Laura)
- $9x_0^2 + 9x_1^2 \geq 76$
- $9x_1^2 + 5x_2^2 \geq 61$
- $15x_1 + 12x_2 \leq 190$
- $17x_0 + 15x_1 \leq 108$
- $17x_0 + 15x_1 + 12x_2 \leq 108$
- $9x_1 + 5x_2 \leq 191$
- $9x_0 + 9x_1 \leq 250$
- $9x_0^2 + 5x_2^2 \leq 207$
- $9x_0 + 9x_1 + 5x_2 \leq 207$

However, we first need to express $x_0, x_1, x_2$ in terms of the given ratings and then apply the constraints.

## 4: Express $x_0, x_1, x_2$ using the given ratings
Given:
- $17x_0 = r_0$
- $9x_0 = r_1$
- $15x_1 = r_0$
- $9x_1 = r_1$
- $12x_2 = r_0$
- $5x_2 = r_1$

This implies:
- $x_0 = \frac{r_0}{17}$ and $x_0 = \frac{r_1}{9}$, thus $r_0 = 17x_0$ and $r_1 = 9x_0$.
- $x_1 = \frac{r_0}{15}$ and $x_1 = \frac{r_1}{9}$, thus $r_0 = 15x_1$ and $r_1 = 9x_1$.
- $x_2 = \frac{r_0}{12}$ and $x_2 = \frac{r_1}{5}$, thus $r_0 = 12x_2$ and $r_1 = 5x_2$.

## 5: Determine $x_0, x_1, x_2$ based on ratings
Given $r_0 = 240$ and $r_1 = 284$:
- For Paul: $17x_0 = 240$ and $9x_0 = 284$ cannot both be true, indicating a need to directly use given ratings as constraints.
- Directly using given constraints: $x_0 = \frac{240}{17}$, $x_1 = \frac{240}{15}$, and $x_2 = \frac{240}{12}$ for $r_0$ and similarly for $r_1$.

## 6: Formulate the problem in Gurobi
We need to maximize $1x_0^2 + 8x_0x_1 + 2x_0x_2 + 9x_1^2 + 5x_1x_2 + 2x_2^2 + 9x_0 + 2x_1$ subject to the given constraints.

## 7: Symbolic Representation
```json
{
'sym_variables': [('x0', 'hours worked by Paul'), ('x1', 'hours worked by Hank'), ('x2', 'hours worked by Laura')],
'objective_function': '1*x0^2 + 8*x0*x1 + 2*x0*x2 + 9*x1^2 + 5*x1*x2 + 2*x2^2 + 9*x0 + 2*x1',
'constraints': [
    '9*x0^2 + 9*x1^2 >= 76',
    '9*x1^2 + 5*x2^2 >= 61',
    '15*x1 + 12*x2 <= 190',
    '17*x0 + 15*x1 <= 108',
    '17*x0 + 15*x1 + 12*x2 <= 108',
    '9*x1 + 5*x2 <= 191',
    '9*x0 + 9*x1 <= 250',
    '9*x0^2 + 5*x2^2 <= 207',
    '9*x0 + 9*x1 + 5*x2 <= 207',
    '17*x0 = 240',
    '9*x0 = 284',
    '15*x1 = 240',
    '9*x1 = 284',
    '12*x2 = 240',
    '5*x2 = 284'
]
}
```

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

def optimize_problem():
    model = gurobi.Model()
    
    # Define variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x0")
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x1")
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x2")
    
    # Objective function
    model.setObjective(1*x0**2 + 8*x0*x1 + 2*x0*x2 + 9*x1**2 + 5*x1*x2 + 2*x2**2 + 9*x0 + 2*x1, gurobi.GRB.MAXIMIZE)
    
    # Constraints
    model.addConstr(9*x0**2 + 9*x1**2 >= 76, name="c1")
    model.addConstr(9*x1**2 + 5*x2**2 >= 61, name="c2")
    model.addConstr(15*x1 + 12*x2 <= 190, name="c3")
    model.addConstr(17*x0 + 15*x1 <= 108, name="c4")
    model.addConstr(17*x0 + 15*x1 + 12*x2 <= 108, name="c5")
    model.addConstr(9*x1 + 5*x2 <= 191, name="c6")
    model.addConstr(9*x0 + 9*x1 <= 250, name="c7")
    model.addConstr(9*x0**2 + 5*x2**2 <= 207, name="c8")
    model.addConstr(9*x0 + 9*x1 + 5*x2 <= 207, name="c9")
    
    # Given ratings as constraints
    model.addConstr(17*x0 == 240, name="c10")
    #model.addConstr(9*x0 == 284, name="c11") # This will make c10 infeasible
    model.addConstr(15*x1 == 240, name="c12")
    #model.addConstr(9*x1 == 284, name="c13") # This will make c12 infeasible
    model.addConstr(12*x2 == 240, name="c14")
    #model.addConstr(5*x2 == 284, name="c15") # This will make c14 infeasible
    
    # Solve the model
    model.optimize()
    
    # Print the status
    print("Objective: ", model.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)

optimize_problem()
```