## Step 1: Define the variables and their corresponding natural language objects
Let's denote the hours worked by Bobby, John, Ringo, Laura, and Jean as $x_0$, $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Formulate the objective function
The objective function to minimize is:
$7.84x_0^2 + 2.19x_0x_1 + 3.28x_0x_2 + 6.15x_0x_3 + 7.03x_0x_4 + 6.23x_1x_2 + 8.13x_2x_3 + 6.08x_2x_4 + 2.24x_3^2 + 9.77x_3x_4 + 3.49x_4^2 + 5.9x_0 + 6.74x_1 + 3.8x_2 + 5.52x_3 + 2.44x_4$

## Step 3: List the constraints
The constraints are:
- $14x_0 \geq 14$ (Bobby's productivity rating)
- $19x_0 \geq 19$ (Bobby's dollar cost per hour)
- $20x_0 \geq 20$ (Bobby's work quality rating)
- $14x_0 \geq 14$ (Bobby's computer competence rating)
- $12x_1 \geq 12$ (John's productivity rating)
- $12x_1 \geq 12$ (John's dollar cost per hour)
- $12x_1 \geq 12$ (John's work quality rating)
- $7x_1 \geq 7$ (John's computer competence rating)
- $8x_2 \geq 8$ (Ringo's productivity rating)
- $2x_2 \geq 2$ (Ringo's dollar cost per hour)
- $12x_2 \geq 12$ (Ringo's work quality rating)
- $12x_2 \geq 12$ (Ringo's computer competence rating)
- $9x_3 \geq 9$ (Laura's productivity rating)
- $8x_3 \geq 8$ (Laura's dollar cost per hour)
- $19x_3 \geq 19$ (Laura's work quality rating)
- $19x_3 \geq 19$ (Laura's computer competence rating)
- $10x_4 \geq 10$ (Jean's productivity rating)
- $20x_4 \geq 20$ (Jean's dollar cost per hour)
- $4x_4 \geq 4$ (Jean's work quality rating)
- $7x_4 \geq 7$ (Jean's computer competence rating)
- $8x_2 + 9x_3 \geq 27$ (Ringo and Laura's productivity rating)
- $12x_1 + 8x_2 \geq 29$ (John and Ringo's productivity rating)
- $14^2x_0^2 + 12^2x_1^2 + 9^2x_3^2 \geq 39$ (Bobby, John, and Laura's productivity rating)
- $14x_0 + 12x_1 + 8x_2 \geq 39$ (Bobby, John, and Ringo's productivity rating)
- $14x_0 + 8x_2 + 9x_3 \geq 39$ (Bobby, Ringo, and Laura's productivity rating)
- $14^2x_0^2 + 8^2x_2^2 + 10^2x_4^2 \geq 39$ (Bobby, Ringo, and Jean's productivity rating)
- $12^2x_1^2 + 8^2x_2^2 + 10^2x_4^2 \geq 39$ (John, Ringo, and Jean's productivity rating)
- $14^2x_0^2 + 12^2x_1^2 + 9^2x_3^2 \geq 51$ (Bobby, John, and Laura's productivity rating)
- $14^2x_0^2 + 12^2x_1^2 + 8^2x_2^2 \geq 51$ (Bobby, John, and Ringo's productivity rating)
- $14x_0 + 8x_2 + 9x_3 \geq 51$ (Bobby, Ringo, and Laura's productivity rating)
- $14x_0 + 8x_2 + 10x_4 \geq 51$ (Bobby, Ringo, and Jean's productivity rating)
- $12x_1 + 8x_2 + 10x_4 \geq 51$ (John, Ringo, and Jean's productivity rating)
- $14x_0 + 12x_1 + 9x_3 \geq 38$ (Bobby, John, and Laura's productivity rating)
- $14x_0 + 12x_1 + 8x_2 \geq 38$ (Bobby, John, and Ringo's productivity rating)
- $14^2x_0^2 + 8^2x_2^2 + 9^2x_3^2 \geq 38$ (Bobby, Ringo, and Laura's productivity rating)
- $14x_0 + 8x_2 + 10x_4 \geq 38$ (Bobby, Ringo, and Jean's productivity rating)
- $12x_1 + 8x_2 + 10x_4 \geq 38$ (John, Ringo, and Jean's productivity rating)
- $14x_0 + 12x_1 + 9x_3 \geq 54$ (Bobby, John, and Laura's productivity rating)
- $14x_0 + 12x_1 + 8x_2 \geq 54$ (Bobby, John, and Ringo's productivity rating)
- $14x_0 + 8x_2 + 9x_3 \geq 54$ (Bobby, Ringo, and Laura's productivity rating)
- $14^2x_0^2 + 8^2x_2^2 + 10^2x_4^2 \geq 54$ (Bobby, Ringo, and Jean's productivity rating)
- $12x_1 + 8x_2 + 10x_4 \geq 54$ (John, Ringo, and Jean's productivity rating)

## 4: Symbolically represent the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Bobby'),
        ('x1', 'hours worked by John'),
        ('x2', 'hours worked by Ringo'),
        ('x3', 'hours worked by Laura'),
        ('x4', 'hours worked by Jean')
    ],
    'objective_function': '7.84*x0^2 + 2.19*x0*x1 + 3.28*x0*x2 + 6.15*x0*x3 + 7.03*x0*x4 + 6.23*x1*x2 + 8.13*x2*x3 + 6.08*x2*x4 + 2.24*x3^2 + 9.77*x3*x4 + 3.49*x4^2 + 5.9*x0 + 6.74*x1 + 3.8*x2 + 5.52*x3 + 2.44*x4',
    'constraints': [
        '14*x0 >= 14',
        '19*x0 >= 19',
        '20*x0 >= 20',
        '14*x0 >= 14',
        '12*x1 >= 12',
        '12*x1 >= 12',
        '12*x1 >= 12',
        '7*x1 >= 7',
        '8*x2 >= 8',
        '2*x2 >= 2',
        '12*x2 >= 12',
        '12*x2 >= 12',
        '9*x3 >= 9',
        '8*x3 >= 8',
        '19*x3 >= 19',
        '19*x3 >= 19',
        '10*x4 >= 10',
        '20*x4 >= 20',
        '4*x4 >= 4',
        '7*x4 >= 7',
        '8*x2 + 9*x3 >= 27',
        '12*x1 + 8*x2 >= 29',
        '14^2*x0^2 + 12^2*x1^2 + 9^2*x3^2 >= 39',
        '14*x0 + 12*x1 + 8*x2 >= 39',
        '14*x0 + 8*x2 + 9*x3 >= 39',
        '14^2*x0^2 + 8^2*x2^2 + 10^2*x4^2 >= 39',
        '12^2*x1^2 + 8^2*x2^2 + 10^2*x4^2 >= 39',
        '14^2*x0^2 + 12^2*x1^2 + 9^2*x3^2 >= 51',
        '14^2*x0^2 + 12^2*x1^2 + 8^2*x2^2 >= 51',
        '14*x0 + 8*x2 + 9*x3 >= 51',
        '14*x0 + 8*x2 + 10*x4 >= 51',
        '12*x1 + 8*x2 + 10*x4 >= 51',
        '14*x0 + 12*x1 + 9*x3 >= 38',
        '14*x0 + 12*x1 + 8*x2 >= 38',
        '14^2*x0^2 + 8^2*x2^2 + 9^2*x3^2 >= 38',
        '14*x0 + 8*x2 + 10*x4 >= 38',
        '12*x1 + 8*x2 + 10*x4 >= 38',
        '14*x0 + 12*x1 + 9*x3 >= 54',
        '14*x0 + 12*x1 + 8*x2 >= 54',
        '14*x0 + 8*x2 + 9*x3 >= 54',
        '14^2*x0^2 + 8^2*x2^2 + 10^2*x4^2 >= 54',
        '12*x1 + 8*x2 + 10*x4 >= 54',
        'x2 >= 0', 'x3 >= 0', 'x4 >= 0', 'x0 >= 0', 'x1 >= 0'
    ]
}
```

## Step 5: Implement the problem in Gurobi
```python
import gurobi

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

# Define the variables
x0 = m.addVar(lb=0, name="x0")
x1 = m.addVar(lb=0, name="x1")
x2 = m.addVar(lb=0, name="x2", vtype=gurobi.GRB.INTEGER)
x3 = m.addVar(lb=0, name="x3", vtype=gurobi.GRB.INTEGER)
x4 = m.addVar(lb=0, name="x4", vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(7.84*x0**2 + 2.19*x0*x1 + 3.28*x0*x2 + 6.15*x0*x3 + 7.03*x0*x4 + 
               6.23*x1*x2 + 8.13*x2*x3 + 6.08*x2*x4 + 2.24*x3**2 + 9.77*x3*x4 + 
               3.49*x4**2 + 5.9*x0 + 6.74*x1 + 3.8*x2 + 5.52*x3 + 2.44*x4, 
               gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(14*x0 >= 14)
m.addConstr(19*x0 >= 19)
m.addConstr(20*x0 >= 20)
m.addConstr(14*x0 >= 14)
m.addConstr(12*x1 >= 12)
m.addConstr(12*x1 >= 12)
m.addConstr(12*x1 >= 12)
m.addConstr(7*x1 >= 7)
m.addConstr(8*x2 >= 8)
m.addConstr(2*x2 >= 2)
m.addConstr(12*x2 >= 12)
m.addConstr(12*x2 >= 12)
m.addConstr(9*x3 >= 9)
m.addConstr(8*x3 >= 8)
m.addConstr(19*x3 >= 19)
m.addConstr(19*x3 >= 19)
m.addConstr(10*x4 >= 10)
m.addConstr(20*x4 >= 20)
m.addConstr(4*x4 >= 4)
m.addConstr(7*x4 >= 7)
m.addConstr(8*x2 + 9*x3 >= 27)
m.addConstr(12*x1 + 8*x2 >= 29)
m.addConstr(14**2*x0**2 + 12**2*x1**2 + 9**2*x3**2 >= 39)
m.addConstr(14*x0 + 12*x1 + 8*x2 >= 39)
m.addConstr(14*x0 + 8*x2 + 9*x3 >= 39)
m.addConstr(14**2*x0**2 + 8**2*x2**2 + 10**2*x4**2 >= 39)
m.addConstr(12**2*x1**2 + 8**2*x2**2 + 10**2*x4**2 >= 39)
m.addConstr(14**2*x0**2 + 12**2*x1**2 + 9**2*x3**2 >= 51)
m.addConstr(14**2*x0**2 + 12**2*x1**2 + 8**2*x2**2 >= 51)
m.addConstr(14*x0 + 8*x2 + 9*x3 >= 51)
m.addConstr(14*x0 + 8*x2 + 10*x4 >= 51)
m.addConstr(12*x1 + 8*x2 + 10*x4 >= 51)
m.addConstr(14*x0 + 12*x1 + 9*x3 >= 38)
m.addConstr(14*x0 + 12*x1 + 8*x2 >= 38)
m.addConstr(14**2*x0**2 + 8**2*x2**2 + 9**2*x3**2 >= 38)
m.addConstr(14*x0 + 8*x2 + 10*x4 >= 38)
m.addConstr(12*x1 + 8*x2 + 10*x4 >= 38)
m.addConstr(14*x0 + 12*x1 + 9*x3 >= 54)
m.addConstr(14*x0 + 12*x1 + 8*x2 >= 54)
m.addConstr(14*x0 + 8*x2 + 9*x3 >= 54)
m.addConstr(14**2*x0**2 + 8**2*x2**2 + 10**2*x4**2 >= 54)
m.addConstr(12*x1 + 8*x2 + 10*x4 >= 54)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.Status.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
    print("x4: ", x4.varValue)
else:
    print("No solution found")
```