## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by George', 'hours worked by Hank', 'hours worked by Peggy', 'hours worked by Ringo', which can be represented symbolically as $x_1, x_2, x_3, x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $6x_1^2 + 7x_1x_2 + 5x_1x_3 + 1x_1x_4 + 3x_2^2 + 9x_2x_3 + 8x_2x_4 + 7x_3^2 + 2x_3x_4 + 4x_4^2 + 6x_1 + 5x_2 + 2x_3 + 2x_4$.

## 3: List the constraints in symbolic notation
The constraints are:
1. $x_1 = 3$ (George's organization score is 3)
2. $x_2 = 11$ (Hank's organization score is 11)
3. $x_3 = 12$ (Peggy's organization score is 12)
4. $x_4 = 13$ (Ringo's organization score is 13)
5. $x_2^2 + x_3^2 \geq 25$ (The total combined organization score from hours worked by Hank squared plus hours worked by Peggy squared has to be 25 at a minimum)
6. $x_2^2 + x_4^2 \geq 17$ (The total combined organization score from hours worked by Hank squared plus hours worked by Ringo squared should be 17 or more)
7. $x_1 + x_2 \geq 14$ (The total combined organization score from hours worked by George plus hours worked by Hank should be no less than 14)
8. $x_1 + x_4 \geq 16$ (The total combined organization score from hours worked by George plus hours worked by Ringo should be no less than 16)
9. $x_1 + x_2 + x_3 + x_4 \geq 16$ (The total combined organization score from hours worked by George plus hours worked by Hank plus hours worked by Peggy plus hours worked by Ringo must be 16 at minimum)
10. $-10x_1 + 4x_4 \geq 0$ (-10 times the number of hours worked by George, plus 4 times the number of hours worked by Ringo has to be no less than zero)

## 4: Define the symbolic representation of the problem
```json
{
'sym_variables': [
    ('x1', 'hours worked by George'), 
    ('x2', 'hours worked by Hank'), 
    ('x3', 'hours worked by Peggy'), 
    ('x4', 'hours worked by Ringo')
],
'objective_function': '6*x1^2 + 7*x1*x2 + 5*x1*x3 + 1*x1*x4 + 3*x2^2 + 9*x2*x3 + 8*x2*x4 + 7*x3^2 + 2*x3*x4 + 4*x4^2 + 6*x1 + 5*x2 + 2*x3 + 2*x4',
'constraints': [
    'x1 = 3', 
    'x2 = 11', 
    'x3 = 12', 
    'x4 = 13', 
    'x2^2 + x3^2 >= 25', 
    'x2^2 + x4^2 >= 17', 
    'x1 + x2 >= 14', 
    'x1 + x4 >= 16', 
    'x1 + x2 + x3 + x4 >= 16', 
    '-10*x1 + 4*x4 >= 0'
]
}
```

## Step 5: Write the Gurobi code for the optimization problem
```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name='x1', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.CONTINUOUS)  # hours worked by George
    x2 = model.addVar(name='x2', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
    x3 = model.addVar(name='x3', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy
    x4 = model.addVar(name='x4', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo

    # Set the objective function
    model.setObjective(6*x1**2 + 7*x1*x2 + 5*x1*x3 + x1*x4 + 3*x2**2 + 9*x2*x3 + 8*x2*x4 + 7*x3**2 + 2*x3*x4 + 4*x4**2 + 6*x1 + 5*x2 + 2*x3 + 2*x4, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(x1 == 3)
    model.addConstr(x2 == 11)
    model.addConstr(x3 == 12)
    model.addConstr(x4 == 13)
    model.addConstr(x2**2 + x3**2 >= 25)
    model.addConstr(x2**2 + x4**2 >= 17)
    model.addConstr(x1 + x2 >= 14)
    model.addConstr(x1 + x4 >= 16)
    model.addConstr(x1 + x2 + x3 + x4 >= 16)
    model.addConstr(-10*x1 + 4*x4 >= 0)

    # Update the model
    model.update()

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('x1: ', x1.varValue)
        print('x2: ', x2.varValue)
        print('x3: ', x3.varValue)
        print('x4: ', x4.varValue)
    else:
        print('The model is infeasible')

solve_optimization_problem()
```