## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Peggy', 'hours worked by Laura', 'hours worked by Paul', 'hours worked by John']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $8.12x_0^2 + 8.48x_0x_2 + 8.94x_0x_3 + 5.51x_1^2 + 2.61x_1x_2 + 9.03x_1x_3 + 9.34x_2^2 + 6.72x_2x_3 + 8.88x_1 + 8.85x_2 + 8.95x_3$.

## 3: List the constraints
The constraints are:
- $0.76x_0 = 0.76$
- $9.72x_0 = 9.72$
- $8.0x_1 = 8.0$
- $1.33x_1 = 1.33$
- $6.02x_2 = 6.02$
- $9.22x_2 = 9.22$
- $1.22x_3 = 1.22$
- $6.98x_3 = 6.98$
- $0.76^2x_0^2 + 6.02^2x_2^2 \geq 18$
- $0.76x_0 + 1.22x_3 \geq 21$
- $0.76x_0 + 8.0x_1 + 6.02x_2 \geq 34$
- $0.76x_0 + 6.02x_2 + 1.22x_3 \geq 34$
- $0.76^2x_0^2 + 8.0^2x_1^2 + 6.02^2x_2^2 \geq 41$
- $0.76x_0 + 6.02x_2 + 1.22x_3 \geq 41$
- $0.76x_0 + 8.0x_1 + 6.02x_2 + 1.22x_3 \geq 41$
- $1.33x_1 + 6.98x_3 \geq 6$
- $9.72^2x_0^2 + 9.22^2x_2^2 \geq 15$
- $9.72x_0 + 1.33x_1 \geq 13$
- $9.72x_0 + 1.33x_1 + 9.22x_2 \geq 12$
- $1.33^2x_1^2 + 9.22^2x_2^2 + 6.98^2x_3^2 \geq 12$
- $9.72^2x_0^2 + 1.33^2x_1^2 + 9.22^2x_2^2 \geq 17$
- $1.33x_1 + 9.22x_2 + 6.98x_3 \geq 17$
- $9.72x_0 + 1.33x_1 + 9.22x_2 + 6.98x_3 \geq 17$
- $-6x_0 + 8x_1 \geq 0$
- $0.76x_0 + 8.0x_1 \leq 148$
- $8.0x_1 + 6.02x_2 + 1.22x_3 \leq 166$
- $0.76^2x_0^2 + 8.0^2x_1^2 + 6.02^2x_2^2 \leq 148$
- $0.76^2x_0^2 + 8.0^2x_1^2 + 1.22^2x_3^2 \leq 115$
- $0.76x_0 + 6.02x_2 + 1.22x_3 \leq 89$
- $9.72x_0 + 1.33x_1 + 9.22x_2 \leq 41$

## 4: Convert the problem into Gurobi code
```python
import gurobi

# Define the model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Peggy
x1 = m.addVar(name="x1", lb=0)  # hours worked by Laura
x2 = m.addVar(name="x2", lb=0)  # hours worked by Paul
x3 = m.addVar(name="x3", lb=0)  # hours worked by John

# Objective function
m.setObjective(8.12*x0**2 + 8.48*x0*x2 + 8.94*x0*x3 + 5.51*x1**2 + 2.61*x1*x2 + 9.03*x1*x3 + 9.34*x2**2 + 6.72*x2*x3 + 8.88*x1 + 8.85*x2 + 8.95*x3, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(0.76*x0 == 0.76)
m.addConstr(9.72*x0 == 9.72)
m.addConstr(8.0*x1 == 8.0)
m.addConstr(1.33*x1 == 1.33)
m.addConstr(6.02*x2 == 6.02)
m.addConstr(9.22*x2 == 9.22)
m.addConstr(1.22*x3 == 1.22)
m.addConstr(6.98*x3 == 6.98)
m.addConstr(0.76**2*x0**2 + 6.02**2*x2**2 >= 18)
m.addConstr(0.76*x0 + 1.22*x3 >= 21)
m.addConstr(0.76*x0 + 8.0*x1 + 6.02*x2 >= 34)
m.addConstr(0.76*x0 + 6.02*x2 + 1.22*x3 >= 34)
m.addConstr(0.76**2*x0**2 + 8.0**2*x1**2 + 6.02**2*x2**2 >= 41)
m.addConstr(0.76*x0 + 6.02*x2 + 1.22*x3 >= 41)
m.addConstr(0.76*x0 + 8.0*x1 + 6.02*x2 + 1.22*x3 >= 41)
m.addConstr(1.33*x1 + 6.98*x3 >= 6)
m.addConstr(9.72**2*x0**2 + 9.22**2*x2**2 >= 15)
m.addConstr(9.72*x0 + 1.33*x1 >= 13)
m.addConstr(9.72*x0 + 1.33*x1 + 9.22*x2 >= 12)
m.addConstr(1.33**2*x1**2 + 9.22**2*x2**2 + 6.98**2*x3**2 >= 12)
m.addConstr(9.72**2*x0**2 + 1.33**2*x1**2 + 9.22**2*x2**2 >= 17)
m.addConstr(1.33*x1 + 9.22*x2 + 6.98*x3 >= 17)
m.addConstr(9.72*x0 + 1.33*x1 + 9.22*x2 + 6.98*x3 >= 17)
m.addConstr(-6*x0 + 8*x1 >= 0)
m.addConstr(0.76*x0 + 8.0*x1 <= 148)
m.addConstr(8.0*x1 + 6.02*x2 + 1.22*x3 <= 166)
m.addConstr(0.76**2*x0**2 + 8.0**2*x1**2 + 6.02**2*x2**2 <= 148)
m.addConstr(0.76**2*x0**2 + 8.0**2*x1**2 + 1.22**2*x3**2 <= 115)
m.addConstr(0.76*x0 + 6.02*x2 + 1.22*x3 <= 89)
m.addConstr(9.72*x0 + 1.33*x1 + 9.22*x2 <= 41)

# 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)
    print("x3: ", x3.varValue)
else:
    print("The model is infeasible")
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Peggy'),
        ('x1', 'hours worked by Laura'),
        ('x2', 'hours worked by Paul'),
        ('x3', 'hours worked by John')
    ],
    'objective_function': '8.12*x0^2 + 8.48*x0*x2 + 8.94*x0*x3 + 5.51*x1^2 + 2.61*x1*x2 + 9.03*x1*x3 + 9.34*x2^2 + 6.72*x2*x3 + 8.88*x1 + 8.85*x2 + 8.95*x3',
    'constraints': [
        '0.76*x0 == 0.76',
        '9.72*x0 == 9.72',
        '8.0*x1 == 8.0',
        '1.33*x1 == 1.33',
        '6.02*x2 == 6.02',
        '9.22*x2 == 9.22',
        '1.22*x3 == 1.22',
        '6.98*x3 == 6.98',
        '0.76^2*x0^2 + 6.02^2*x2^2 >= 18',
        '0.76*x0 + 1.22*x3 >= 21',
        '0.76*x0 + 8.0*x1 + 6.02*x2 >= 34',
        '0.76*x0 + 6.02*x2 + 1.22*x3 >= 34',
        '0.76^2*x0^2 + 8.0^2*x1^2 + 6.02^2*x2^2 >= 41',
        '0.76*x0 + 6.02*x2 + 1.22*x3 >= 41',
        '0.76*x0 + 8.0*x1 + 6.02*x2 + 1.22*x3 >= 41',
        '1.33*x1 + 6.98*x3 >= 6',
        '9.72^2*x0^2 + 9.22^2*x2^2 >= 15',
        '9.72*x0 + 1.33*x1 >= 13',
        '9.72*x0 + 1.33*x1 + 9.22*x2 >= 12',
        '1.33^2*x1^2 + 9.22^2*x2^2 + 6.98^2*x3^2 >= 12',
        '9.72^2*x0^2 + 1.33^2*x1^2 + 9.22^2*x2^2 >= 17',
        '1.33*x1 + 9.22*x2 + 6.98*x3 >= 17',
        '9.72*x0 + 1.33*x1 + 9.22*x2 + 6.98*x3 >= 17',
        '-6*x0 + 8*x1 >= 0',
        '0.76*x0 + 8.0*x1 <= 148',
        '8.0*x1 + 6.02*x2 + 1.22*x3 <= 166',
        '0.76^2*x0^2 + 8.0^2*x1^2 + 6.02^2*x2^2 <= 148',
        '0.76^2*x0^2 + 8.0^2*x1^2 + 1.22^2*x3^2 <= 115',
        '0.76*x0 + 6.02*x2 + 1.22*x3 <= 89',
        '9.72*x0 + 1.33*x1 + 9.22*x2 <= 41'
    ]
}
```