## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Mary' and 'hours worked by Laura'. Let's denote 'hours worked by Mary' as $x_1$ and 'hours worked by Laura' as $x_2$. The objective function to minimize is $4x_1^2 + 7x_1$. The constraints are:
- $2x_1 + 20x_2 \geq 35$
- $4x_1 + 11x_2 \geq 78$
- $5x_1 + 5x_2 \geq 50$
- $7x_1^2 + 18x_2^2 \geq 31$
- $11x_1 + x_2 \geq 38$
- $2x_1 - 3x_2 \geq 0$
- $2^2x_1^2 + 20^2x_2^2 \leq 45$
- $4x_1 + 11x_2 \leq 122$
- $5x_1 + 5x_2 \leq 116$
- $7x_1^2 + 18x_2^2 \leq 95$
- $11x_1 + x_2 \leq 77$

## Step 2: Convert the problem into a Gurobi-compatible format
We need to define the variables, the objective function, and the constraints in a format that Gurobi can understand.

## 3: Write the Gurobi code
```python
import gurobi as gp

# Define the model
model = gp.Model()

# Define the variables
x1 = model.addVar(name="hours_worked_by_Mary", lb=0)  # hours worked by Mary
x2 = model.addVar(name="hours_worked_by_Laura", lb=0)  # hours worked by Laura

# Define the objective function
model.setObjective(4 * x1**2 + 7 * x1, gp.GRB.MINIMIZE)

# Define the constraints
model.addConstr(2 * x1 + 20 * x2 >= 35, name="dollar_cost_per_hour")
model.addConstr(4 * x1 + 11 * x2 >= 78, name="computer_competence_rating")
model.addConstr(5 * x1 + 5 * x2 >= 50, name="productivity_rating")
model.addConstr(7 * x1**2 + 18 * x2**2 >= 31, name="paperwork_competence_rating")
model.addConstr(11 * x1 + x2 >= 38, name="work_quality_rating")
model.addConstr(2 * x1 - 3 * x2 >= 0, name="hourly_work_constraint")
model.addConstr(2**2 * x1**2 + 20**2 * x2**2 <= 45, name="dollar_cost_per_hour_squared")
model.addConstr(4 * x1 + 11 * x2 <= 122, name="computer_competence_rating_upper_bound")
model.addConstr(5 * x1 + 5 * x2 <= 116, name="productivity_rating_upper_bound")
model.addConstr(7 * x1**2 + 18 * x2**2 <= 95, name="paperwork_competence_rating_upper_bound")
model.addConstr(11 * x1 + x2 <= 77, name="work_quality_rating_upper_bound")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Mary: {x1.varValue}")
    print(f"Hours worked by Laura: {x2.varValue}")
    print(f"Objective function value: {model.objVal}")
else:
    print("No optimal solution found.")
```

## 4: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Mary'), ('x2', 'hours worked by Laura')],
    'objective_function': '4*x1^2 + 7*x1',
    'constraints': [
        '2*x1 + 20*x2 >= 35',
        '4*x1 + 11*x2 >= 78',
        '5*x1 + 5*x2 >= 50',
        '7*x1^2 + 18*x2^2 >= 31',
        '11*x1 + x2 >= 38',
        '2*x1 - 3*x2 >= 0',
        '2^2*x1^2 + 20^2*x2^2 <= 45',
        '4*x1 + 11*x2 <= 122',
        '5*x1 + 5*x2 <= 116',
        '7*x1^2 + 18*x2^2 <= 95',
        '11*x1 + x2 <= 77'
    ]
}
```