## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Paul', 'hours worked by Peggy', 'hours worked by Bobby', 'hours worked by Mary'].
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 $7x_0x_1 + 6x_1x_2 + 6x_3^2 + x_0 + 7x_3$.

## 3: List the constraints
The constraints are:
- $x_0 \geq 0$, $x_1 \geq 0$, $x_2 \geq 0$, $x_3 \geq 0$ (non-negativity constraints, though not explicitly stated, implied for hours worked)
- $15.1x_0 \leq 173$
- $9.85x_0 \leq 256$
- $18.65x_1 \leq 173$
- $17.94x_1 \leq 256$
- $7.75x_2 \leq 173$
- $7.48x_2 \leq 256$
- $0.8x_3 \leq 173$
- $1.13x_3 \leq 256$
- $15.1^2x_0^2 + 18.65^2x_1^2 \geq 26$
- $15.1x_0 + 7.75x_2 \geq 26$
- $7.75x_2 + 0.8x_3 \geq 15$
- $18.65x_1 + 0.8x_3 \geq 20$
- $15.1x_0 + 18.65x_1 + 0.8x_3 \geq 28$
- $15.1x_0 + 18.65x_1 + 7.75x_2 + 0.8x_3 \geq 28$
- $9.85x_0 + 17.94x_1 \geq 25$
- $9.85^2x_0^2 + 17.94^2x_1^2 + 7.48^2x_2^2 \geq 53$
- $9.85x_0 + 7.48x_2 + 1.13x_3 \geq 53$
- $9.85^2x_0^2 + 17.94^2x_1^2 + 1.13^2x_3^2 \geq 53$
- $9.85^2x_0^2 + 17.94^2x_1^2 + 7.48^2x_2^2 \geq 45$
- $9.85x_0 + 7.48x_2 + 1.13x_3 \geq 45$
- $9.85x_0 + 17.94x_1 + 1.13x_3 \geq 45$
- $9.85x_0 + 17.94x_1 + 7.48x_2 \geq 32$
- $9.85x_0 + 7.48x_2 + 1.13x_3 \geq 32$
- $9.85^2x_0^2 + 17.94^2x_1^2 + 1.13^2x_3^2 \geq 32$
- $9.85x_0 + 17.94x_1 + 7.48x_2 + 1.13x_3 \geq 32$
- $-3x_0 + 7x_2 \geq 0$
- $18.65x_1 + 7.75x_2 + 0.8x_3 \leq 74$
- $15.1x_0 + 18.65x_1 + 0.8x_3 \leq 137$
- $15.1^2x_0^2 + 7.75^2x_2^2 + 0.8^2x_3^2 \leq 103$
- $17.94x_1 + 7.48x_2 \leq 245$
- $9.85x_0 + 1.13x_3 \leq 226$
- $9.85x_0 + 17.94x_1 + 1.13x_3 \leq 206$
- $9.85x_0 + 17.94x_1 + 7.48x_2 \leq 127$
- $9.85x_0 + 7.48x_2 + 1.13x_3 \leq 150$

## 4: Specify variable types
- $x_0$ is continuous
- $x_1$ is continuous
- $x_2$ is integer
- $x_3$ is continuous

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

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

# Define variables
x0 = m.addVar(lb=0, name="x0", vtype=gurobi.GRB.CONTINUOUS)  # hours worked by Paul
x1 = m.addVar(lb=0, name="x1", vtype=gurobi.GRB.CONTINUOUS)  # hours worked by Peggy
x2 = m.addVar(lb=0, name="x2", vtype=gurobi.GRB.INTEGER)    # hours worked by Bobby
x3 = m.addVar(lb=0, name="x3", vtype=gurobi.GRB.CONTINUOUS)  # hours worked by Mary

# Objective function
m.setObjective(7*x0*x1 + 6*x1*x2 + 6*x3**2 + x0 + 7*x3, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(15.1 * x0 <= 173)
m.addConstr(9.85 * x0 <= 256)
m.addConstr(18.65 * x1 <= 173)
m.addConstr(17.94 * x1 <= 256)
m.addConstr(7.75 * x2 <= 173)
m.addConstr(7.48 * x2 <= 256)
m.addConstr(0.8 * x3 <= 173)
m.addConstr(1.13 * x3 <= 256)
m.addConstr(15.1**2 * x0**2 + 18.65**2 * x1**2 >= 26)
m.addConstr(15.1 * x0 + 7.75 * x2 >= 26)
m.addConstr(7.75 * x2 + 0.8 * x3 >= 15)
m.addConstr(18.65 * x1 + 0.8 * x3 >= 20)
m.addConstr(15.1 * x0 + 18.65 * x1 + 0.8 * x3 >= 28)
m.addConstr(15.1 * x0 + 18.65 * x1 + 7.75 * x2 + 0.8 * x3 >= 28)
m.addConstr(9.85 * x0 + 17.94 * x1 >= 25)
m.addConstr(9.85**2 * x0**2 + 17.94**2 * x1**2 + 7.48**2 * x2**2 >= 53)
m.addConstr(9.85 * x0 + 7.48 * x2 + 1.13 * x3 >= 53)
m.addConstr(9.85**2 * x0**2 + 17.94**2 * x1**2 + 1.13**2 * x3**2 >= 53)
m.addConstr(9.85**2 * x0**2 + 17.94**2 * x1**2 + 7.48**2 * x2**2 >= 45)
m.addConstr(9.85 * x0 + 7.48 * x2 + 1.13 * x3 >= 45)
m.addConstr(9.85 * x0 + 17.94 * x1 + 1.13 * x3 >= 45)
m.addConstr(9.85 * x0 + 17.94 * x1 + 7.48 * x2 >= 32)
m.addConstr(9.85 * x0 + 7.48 * x2 + 1.13 * x3 >= 32)
m.addConstr(9.85**2 * x0**2 + 17.94**2 * x1**2 + 1.13**2 * x3**2 >= 32)
m.addConstr(9.85 * x0 + 17.94 * x1 + 7.48 * x2 + 1.13 * x3 >= 32)
m.addConstr(-3 * x0 + 7 * x2 >= 0)
m.addConstr(18.65 * x1 + 7.75 * x2 + 0.8 * x3 <= 74)
m.addConstr(15.1 * x0 + 18.65 * x1 + 0.8 * x3 <= 137)
m.addConstr(15.1**2 * x0**2 + 7.75**2 * x2**2 + 0.8**2 * x3**2 <= 103)
m.addConstr(17.94 * x1 + 7.48 * x2 <= 245)
m.addConstr(9.85 * x0 + 1.13 * x3 <= 226)
m.addConstr(9.85 * x0 + 17.94 * x1 + 1.13 * x3 <= 206)
m.addConstr(9.85 * x0 + 17.94 * x1 + 7.48 * x2 <= 127)
m.addConstr(9.85 * x0 + 7.48 * x2 + 1.13 * x3 <= 150)

# Optimize
m.optimize()

# Print 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("No solution found")
```

## Step 6: Symbolic representation
```json
{
    'sym_variables': [
        ['x0', 'hours worked by Paul'], 
        ['x1', 'hours worked by Peggy'], 
        ['x2', 'hours worked by Bobby'], 
        ['x3', 'hours worked by Mary']
    ], 
    'objective_function': '7*x0*x1 + 6*x1*x2 + 6*x3^2 + x0 + 7*x3', 
    'constraints': [
        '15.1*x0 <= 173',
        '9.85*x0 <= 256',
        '18.65*x1 <= 173',
        '17.94*x1 <= 256',
        '7.75*x2 <= 173',
        '7.48*x2 <= 256',
        '0.8*x3 <= 173',
        '1.13*x3 <= 256',
        '15.1^2*x0^2 + 18.65^2*x1^2 >= 26',
        '15.1*x0 + 7.75*x2 >= 26',
        '7.75*x2 + 0.8*x3 >= 15',
        '18.65*x1 + 0.8*x3 >= 20',
        '15.1*x0 + 18.65*x1 + 0.8*x3 >= 28',
        '15.1*x0 + 18.65*x1 + 7.75*x2 + 0.8*x3 >= 28',
        '9.85*x0 + 17.94*x1 >= 25',
        '9.85^2*x0^2 + 17.94^2*x1^2 + 7.48^2*x2^2 >= 53',
        '9.85*x0 + 7.48*x2 + 1.13*x3 >= 53',
        '9.85^2*x0^2 + 17.94^2*x1^2 + 1.13^2*x3^2 >= 53',
        '9.85^2*x0^2 + 17.94^2*x1^2 + 7.48^2*x2^2 >= 45',
        '9.85*x0 + 7.48*x2 + 1.13*x3 >= 45',
        '9.85*x0 + 17.94*x1 + 1.13*x3 >= 45',
        '9.85*x0 + 17.94*x1 + 7.48*x2 >= 32',
        '9.85*x0 + 7.48*x2 + 1.13*x3 >= 32',
        '9.85^2*x0^2 + 17.94^2*x1^2 + 1.13^2*x3^2 >= 32',
        '9.85*x0 + 17.94*x1 + 7.48*x2 + 1.13*x3 >= 32',
        '-3*x0 + 7*x2 >= 0',
        '18.65*x1 + 7.75*x2 + 0.8*x3 <= 74',
        '15.1*x0 + 18.65*x1 + 0.8*x3 <= 137',
        '15.1^2*x0^2 + 7.75^2*x2^2 + 0.8^2*x3^2 <= 103',
        '17.94*x1 + 7.48*x2 <= 245',
        '9.85*x0 + 1.13*x3 <= 226',
        '9.85*x0 + 17.94*x1 + 1.13*x3 <= 206',
        '9.85*x0 + 17.94*x1 + 7.48*x2 <= 127',
        '9.85*x0 + 7.48*x2 + 1.13*x3 <= 150'
    ]
}
```