## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Ringo' and 'hours worked by Laura'. Let's denote 'hours worked by Ringo' as $x_1$ and 'hours worked by Laura' as $x_2$. The objective function to maximize is $4x_1^2 + 5x_1x_2 + 8x_2^2 + 2x_1 + x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
- $1.93x_1 \leq 35$
- $0.18x_1 \leq 101$
- $1.96x_1 \leq 116$
- $2.25x_1 \leq 26$
- $2.47x_2 \leq 35$
- $2.89x_2 \leq 101$
- $0.25x_2 \leq 116$
- $0.77x_2 \leq 26$
- $1.93x_1^2 + 2.47x_2^2 \geq 16$
- $0.18x_1^2 + 2.89x_2^2 \geq 21$
- $1.96x_1^2 + 0.25x_2^2 \geq 25$
- $2.25x_1 + 0.77x_2 \geq 6$
- $-4x_1 + 10x_2 \geq 0$
- $1.93x_1^2 + 2.47x_2^2 \leq 25$
- $1.93x_1 + 2.47x_2 \leq 25$
- $0.18x_1^2 + 2.89x_2^2 \leq 92$
- $0.18x_1 + 2.89x_2 \leq 92$
- $1.96x_1^2 + 0.25x_2^2 \leq 109$
- $1.96x_1 + 0.25x_2 \leq 109$
- $2.25x_1 + 0.77x_2 \leq 24$
- $x_1$ is an integer
- $x_2$ can be a real number

## 3: Convert the problem into Gurobi code
We will use Gurobi's Python API to model and solve this problem.

```python
import gurobi as gp

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Ringo", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="hours_worked_by_Laura")

# Define the objective function
m.setObjective(4*x1**2 + 5*x1*x2 + 8*x2**2 + 2*x1 + x2, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(1.93*x1 <= 35)
m.addConstr(0.18*x1 <= 101)
m.addConstr(1.96*x1 <= 116)
m.addConstr(2.25*x1 <= 26)
m.addConstr(2.47*x2 <= 35)
m.addConstr(2.89*x2 <= 101)
m.addConstr(0.25*x2 <= 116)
m.addConstr(0.77*x2 <= 26)
m.addConstr(1.93*x1**2 + 2.47*x2**2 >= 16)
m.addConstr(0.18*x1**2 + 2.89*x2**2 >= 21)
m.addConstr(1.96*x1**2 + 0.25*x2**2 >= 25)
m.addConstr(2.25*x1 + 0.77*x2 >= 6)
m.addConstr(-4*x1 + 10*x2 >= 0)
m.addConstr(1.93*x1**2 + 2.47*x2**2 <= 25)
m.addConstr(1.93*x1 + 2.47*x2 <= 25)
m.addConstr(0.18*x1**2 + 2.89*x2**2 <= 92)
m.addConstr(0.18*x1 + 2.89*x2 <= 92)
m.addConstr(1.96*x1**2 + 0.25*x2**2 <= 109)
m.addConstr(1.96*x1 + 0.25*x2 <= 109)
m.addConstr(2.25*x1 + 0.77*x2 <= 24)

# Optimize the model
m.optimize()

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

## 4: Symbolic representation of the problem
The symbolic representation is as follows:

```json
{
    'sym_variables': [('x1', 'hours worked by Ringo'), ('x2', 'hours worked by Laura')],
    'objective_function': '4*x1^2 + 5*x1*x2 + 8*x2^2 + 2*x1 + x2',
    'constraints': [
        '1.93*x1 <= 35',
        '0.18*x1 <= 101',
        '1.96*x1 <= 116',
        '2.25*x1 <= 26',
        '2.47*x2 <= 35',
        '2.89*x2 <= 101',
        '0.25*x2 <= 116',
        '0.77*x2 <= 26',
        '1.93*x1^2 + 2.47*x2^2 >= 16',
        '0.18*x1^2 + 2.89*x2^2 >= 21',
        '1.96*x1^2 + 0.25*x2^2 >= 25',
        '2.25*x1 + 0.77*x2 >= 6',
        '-4*x1 + 10*x2 >= 0',
        '1.93*x1^2 + 2.47*x2^2 <= 25',
        '1.93*x1 + 2.47*x2 <= 25',
        '0.18*x1^2 + 2.89*x2^2 <= 92',
        '0.18*x1 + 2.89*x2 <= 92',
        '1.96*x1^2 + 0.25*x2^2 <= 109',
        '1.96*x1 + 0.25*x2 <= 109',
        '2.25*x1 + 0.77*x2 <= 24'
    ]
}
```