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

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_0$ : hours worked by Jean
- $x_1$ : hours worked by Peggy
- $x_2$ : hours worked by Hank

## 3: Define the objective function in symbolic notation
The objective function to minimize is $4x_0^2 + 7x_0x_1 + x_2^2 + 4x_0$.

## 4: List the constraints in symbolic notation
The constraints are:
- $7x_0 \geq 7$ (Jean's organization score is 7)
- $13x_0 \geq 13$ (Jean has a paperwork competence rating of 13)
- $17x_0 \geq 17$ (Jean has a productivity rating of 17)
- $7x_0 \geq 7$ (Jean has a computer competence rating of 7)
- $17x_1 \geq 17$ (Peggy has a organization score of 17)
- $19x_1 \geq 19$ (Peggy has a paperwork competence rating of 19)
- $18x_1 \geq 18$ (Peggy has a productivity rating of 18)
- $3x_1 \geq 3$ (Peggy's computer competence rating is 3)
- $17x_2 \geq 17$ (Hank has a organization score of 17)
- $9x_2 \geq 9$ (Hank has a paperwork competence rating of 9)
- $13x_2 \geq 13$ (Hank's productivity rating is 13)
- $17x_2 \geq 17$ (Hank has a computer competence rating of 17)
- $7x_0 + 17x_2 \geq 21$ (The total combined organization score from hours worked by Jean plus hours worked by Hank has to be 21 or more)
- $17x_1 + 17x_2 \geq 29$ (The total combined organization score from hours worked by Peggy plus hours worked by Hank must be at least 29)
- $7x_0^2 + 17x_1^2 + 17x_2^2 \geq 37$ (The total combined organization score from hours worked by Jean squared plus hours worked by Peggy squared plus hours worked by Hank squared must be 37 at minimum)
- $7x_0 + 17x_1 + 17x_2 \geq 37$ (The total combined organization score from hours worked by Jean, hours worked by Peggy, and hours worked by Hank must be 37 at a minimum)
- $19x_1^2 + 9x_2^2 \geq 14$ (The total combined paperwork competence rating from hours worked by Peggy squared plus hours worked by Hank squared must be equal to or greater than 14)
- $13x_0 + 9x_2 \geq 25$ (The total combined paperwork competence rating from hours worked by Jean, and hours worked by Hank has to be 25 at a minimum)
- $13x_0 + 19x_1 + 9x_2 \geq 25$ (The total combined paperwork competence rating from hours worked by Jean, hours worked by Peggy, and hours worked by Hank has to be 25 at minimum)
- $18x_1 + 13x_2 \geq 36$ (The total combined productivity rating from hours worked by Peggy, and hours worked by Hank should be as much or more than 36)
- $17x_0^2 + 18x_1^2 \geq 59$ (The total combined productivity rating from hours worked by Jean squared, and hours worked by Peggy squared has to be equal to or greater than 59)
- $17x_0^2 + 13x_2^2 \geq 44$ (The total combined productivity rating from hours worked by Jean squared plus hours worked by Hank squared should be 44 at minimum)
- $17x_0 + 18x_1 + 13x_2 \geq 55$ (The total combined productivity rating from hours worked by Jean plus hours worked by Peggy plus hours worked by Hank must be at least 55)
- $7x_0 + 3x_1 + 17x_2 \geq 59$ (The total combined computer competence rating from hours worked by Jean plus hours worked by Hank has to be at minimum 59)
- $7x_0 + 3x_1 \geq 56$ (The total combined computer competence rating from hours worked by Jean, and hours worked by Peggy has to be 56 at minimum)
- $7x_0 + 3x_1 + 17x_2 \geq 50$ (The total combined computer competence rating from hours worked by Jean, hours worked by Peggy, and hours worked by Hank has to be 50 or more)
- $-x_0 + 5x_2 \geq 0$ ( $-1 times the number of hours worked by Jean, plus 5 times the number of hours worked by Hank must be at least zero)
- $7x_0^2 + 13x_2^2 \leq 95$ (The total combined organization score from hours worked by Jean squared plus hours worked by Hank squared has to be 95 at maximum)
- $17x_1 + 17x_2 \leq 68$ (The total combined organization score from hours worked by Peggy, and hours worked by Hank should be 68 or less)

## 5: Determine the variable types
- $x_0$ : Continuous
- $x_1$ : Integer
- $x_2$ : Continuous

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

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

# Define the variables
x0 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x0")  # hours worked by Jean
x1 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER, name="x1")  # hours worked by Peggy
x2 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x2")  # hours worked by Hank

# Objective function
m.setObjective(4*x0**2 + 7*x0*x1 + x2**2 + 4*x0, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(7*x0 >= 7, name="c1")
m.addConstr(13*x0 >= 13, name="c2")
m.addConstr(17*x0 >= 17, name="c3")
m.addConstr(7*x0 >= 7, name="c4")
m.addConstr(17*x1 >= 17, name="c5")
m.addConstr(19*x1 >= 19, name="c6")
m.addConstr(18*x1 >= 18, name="c7")
m.addConstr(3*x1 >= 3, name="c8")
m.addConstr(17*x2 >= 17, name="c9")
m.addConstr(9*x2 >= 9, name="c10")
m.addConstr(13*x2 >= 13, name="c11")
m.addConstr(17*x2 >= 17, name="c12")
m.addConstr(7*x0 + 17*x2 >= 21, name="c13")
m.addConstr(17*x1 + 17*x2 >= 29, name="c14")
m.addConstr(7*x0**2 + 17*x1**2 + 17*x2**2 >= 37, name="c15")
m.addConstr(7*x0 + 17*x1 + 17*x2 >= 37, name="c16")
m.addConstr(19*x1**2 + 9*x2**2 >= 14, name="c17")
m.addConstr(13*x0 + 9*x2 >= 25, name="c18")
m.addConstr(13*x0 + 19*x1 + 9*x2 >= 25, name="c19")
m.addConstr(18*x1 + 13*x2 >= 36, name="c20")
m.addConstr(17*x0**2 + 18*x1**2 >= 59, name="c21")
m.addConstr(17*x0**2 + 13*x2**2 >= 44, name="c22")
m.addConstr(17*x0 + 18*x1 + 13*x2 >= 55, name="c23")
m.addConstr(7*x0 + 3*x1 + 17*x2 >= 59, name="c24")
m.addConstr(7*x0 + 3*x1 >= 56, name="c25")
m.addConstr(7*x0 + 3*x1 + 17*x2 >= 50, name="c26")
m.addConstr(-x0 + 5*x2 >= 0, name="c27")
m.addConstr(7*x0**2 + 13*x2**2 <= 95, name="c28")
m.addConstr(17*x1 + 17*x2 <= 68, name="c29")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 7: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Jean'), 
        ('x1', 'hours worked by Peggy'), 
        ('x2', 'hours worked by Hank')
    ], 
    'objective_function': '4*x0^2 + 7*x0*x1 + x2^2 + 4*x0', 
    'constraints': [
        '7*x0 >= 7', 
        '13*x0 >= 13', 
        '17*x0 >= 17', 
        '7*x0 >= 7', 
        '17*x1 >= 17', 
        '19*x1 >= 19', 
        '18*x1 >= 18', 
        '3*x1 >= 3', 
        '17*x2 >= 17', 
        '9*x2 >= 9', 
        '13*x2 >= 13', 
        '17*x2 >= 17', 
        '7*x0 + 17*x2 >= 21', 
        '17*x1 + 17*x2 >= 29', 
        '7*x0^2 + 17*x1^2 + 17*x2^2 >= 37', 
        '7*x0 + 17*x1 + 17*x2 >= 37', 
        '19*x1^2 + 9*x2^2 >= 14', 
        '13*x0 + 9*x2 >= 25', 
        '13*x0 + 19*x1 + 9*x2 >= 25', 
        '18*x1 + 13*x2 >= 36', 
        '17*x0^2 + 18*x1^2 >= 59', 
        '17*x0^2 + 13*x2^2 >= 44', 
        '17*x0 + 18*x1 + 13*x2 >= 55', 
        '7*x0 + 3*x1 + 17*x2 >= 59', 
        '7*x0 + 3*x1 >= 56', 
        '7*x0 + 3*x1 + 17*x2 >= 50', 
        '-x0 + 5*x2 >= 0', 
        '7*x0^2 + 13*x2^2 <= 95', 
        '17*x1 + 17*x2 <= 68'
    ]
}
```