## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Jean' and 'hours worked by Bill', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $3x_1^2 + 4x_2^2 + x_1$.

## Step 3: List the constraints
The constraints are:
- $18x_1 + 23x_2 \geq 75$
- $12x_1^2 + 11x_2^2 \geq 62$
- $12x_1 + 11x_2 \geq 62$
- $4x_1 - 8x_2 \geq 0$
- $18x_1 + 23x_2 \leq 89$
- $12x_1 + 11x_2 \leq 124$
- $x_1$ is a non-negative real number (not explicitly stated but implied for hours worked)
- $x_2$ is a non-negative integer

## 4: Convert the problem into a Gurobi-compatible format
We need to express the problem in a way that Gurobi can understand. This involves creating a model, adding variables with their respective properties, setting the objective function, and adding constraints.

## 5: Write the Gurobi code
```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(lb=0, name="hours_worked_by_Jean")  # non-negative real
x2 = model.addVar(lb=0, type=gurobi.GRB.INTEGER, name="hours_worked_by_Bill")  # non-negative integer

# Set the objective function
model.setObjective(3*x1**2 + 4*x2**2 + x1, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(18*x1 + 23*x2 >= 75, name="computer_competence_rating_min")
model.addConstr(12*x1**2 + 11*x2**2 >= 62, name="dollar_cost_per_hour_min")
model.addConstr(12*x1 + 11*x2 >= 62, name="dollar_cost_per_hour_min_linear")
model.addConstr(4*x1 - 8*x2 >= 0, name="jean_bill_hours_constraint")
model.addConstr(18*x1 + 23*x2 <= 89, name="computer_competence_rating_max")
model.addConstr(12*x1 + 11*x2 <= 124, name="dollar_cost_per_hour_max")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Jean: {x1.varValue}")
    print(f"Hours worked by Bill: {x2.varValue}")
else:
    print("No optimal solution found.")
```

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'hours worked by Jean'), ('x2', 'hours worked by Bill')],
    'objective_function': '3*x1^2 + 4*x2^2 + x1',
    'constraints': [
        '18*x1 + 23*x2 >= 75',
        '12*x1^2 + 11*x2^2 >= 62',
        '12*x1 + 11*x2 >= 62',
        '4*x1 - 8*x2 >= 0',
        '18*x1 + 23*x2 <= 89',
        '12*x1 + 11*x2 <= 124'
    ]
}
```