## Step 1: Define the optimization problem
The problem is to minimize the objective function $4 \times (\text{hours worked by Mary})^2 + 7 \times (\text{hours worked by Mary})$, subject to various constraints on the hours worked by Mary and Laura, and their combined ratings.

## 2: List the constraints
The constraints are:
- Mary's dollar cost per hour is 2.
- Mary's computer competence rating is 4.
- Mary has a productivity rating of 5.
- Mary has a paperwork competence rating of 7.
- Mary's work quality rating is 11.
- Laura has a dollar cost per hour of 20.
- Laura has a computer competence rating of 11.
- Laura has a productivity rating of 5.
- Laura's paperwork competence rating is 18.
- Laura has a work quality rating of 1.
- $2 \times (\text{hours worked by Mary}) + 20 \times (\text{hours worked by Laura}) \geq 35$
- $4 \times (\text{hours worked by Mary}) + 11 \times (\text{hours worked by Laura}) \geq 78$
- $5 \times (\text{hours worked by Mary}) + 5 \times (\text{hours worked by Laura}) \geq 50$
- $7 \times (\text{hours worked by Mary}) + 18 \times (\text{hours worked by Laura}) \geq 31$
- $11 \times (\text{hours worked by Mary}) + 1 \times (\text{hours worked by Laura}) \geq 38$
- $2 \times (\text{hours worked by Mary}) - 3 \times (\text{hours worked by Laura}) \geq 0$
- $2^2 \times (\text{hours worked by Mary})^2 + 20^2 \times (\text{hours worked by Laura})^2 \leq 45$
- $4 \times (\text{hours worked by Mary}) + 11 \times (\text{hours worked by Laura}) \leq 122$
- $5 \times (\text{hours worked by Mary}) + 5 \times (\text{hours worked by Laura}) \leq 116$
- $7^2 \times (\text{hours worked by Mary})^2 + 18^2 \times (\text{hours worked by Laura})^2 \leq 95$
- $11 \times (\text{hours worked by Mary}) + 1 \times (\text{hours worked by Laura}) \leq 77$

## 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
hours_worked_by_Mary = m.addVar(name="hours_worked_by_Mary", lb=0)
hours_worked_by_Laura = m.addVar(name="hours_worked_by_Laura", lb=0)

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

# Add constraints
m.addConstr(2 * hours_worked_by_Mary + 20 * hours_worked_by_Laura >= 35)
m.addConstr(4 * hours_worked_by_Mary + 11 * hours_worked_by_Laura >= 78)
m.addConstr(5 * hours_worked_by_Mary + 5 * hours_worked_by_Laura >= 50)
m.addConstr(7 * hours_worked_by_Mary + 18 * hours_worked_by_Laura >= 31)
m.addConstr(11 * hours_worked_by_Mary + hours_worked_by_Laura >= 38)
m.addConstr(2 * hours_worked_by_Mary - 3 * hours_worked_by_Laura >= 0)

m.addConstr(2**2 * hours_worked_by_Mary**2 + 20**2 * hours_worked_by_Laura**2 <= 45)
m.addConstr(4 * hours_worked_by_Mary + 11 * hours_worked_by_Laura <= 122)
m.addConstr(5 * hours_worked_by_Mary + 5 * hours_worked_by_Laura <= 116)
m.addConstr(7**2 * hours_worked_by_Mary**2 + 18**2 * hours_worked_by_Laura**2 <= 95)
m.addConstr(11 * hours_worked_by_Mary + hours_worked_by_Laura <= 77)

# Solve the model
m.optimize()

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