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

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $7.22x_1^2 + 4.56x_1x_2 + 9.3x_1 + 7.44x_2$.

## Step 3: List the constraints in symbolic notation
The constraints are:
1. $5x_1 \geq 5$ (Bobby's productivity rating is 5, but this seems to be an equality, so $5x_1 = 5$ or simply $x_1 \geq 1$ and $x_1 \leq 1$ for equality, however, it is more logical to consider it as $x_1 = 1$ for the rating to be exactly 5),
2. $3x_1 \geq 3$ (Bobby's paperwork competence rating is 3, similarly, $x_1 = 1$),
3. $14x_2 \geq 14$ (Hank's productivity rating is 14, so $x_2 = 1$),
4. $4x_2 \geq 4$ (Hank's paperwork competence rating is 4, so $x_2 = 1$),
5. $5x_1 + 14x_2 \geq 21$,
6. $3x_1 + 4x_2 \geq 20$,
7. $6x_1 - 6x_2 \geq 0$,
8. $x_1^2 + x_2^2 \leq 114$ is not correct based on the problem description. The problem mentions 'The total combined productivity rating from hours worked by Bobby squared plus hours worked by Hank squared has to be 114 at maximum.' which translates to $5^2x_1^2 + 14^2x_2^2 \leq 114$,
9. $3x_1 + 4x_2 \leq 49$.

## 4: Correct and Simplify Constraints
Correcting and simplifying:
- From 1 and 2, we actually have $x_1 = 1$ (as both imply $x_1 = 1$ for their respective ratings to be exactly as stated),
- From 3 and 4, $x_2 = 1$ (as both imply $x_2 = 1$).

## 5: Formulate the Symbolic Representation
Given the corrections, the symbolic representation simplifies significantly because $x_1 = 1$ and $x_2 = 1$ satisfy the first four constraints directly.

## 6: Create the Gurobi Model
We will create a Gurobi model with the given objective function and constraints.

## 7: Write the Gurobi Code
```python
import gurobi

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Bobby", lb=0, ub=None, vtype=gurobi.GRB.CONTINUOUS)
x2 = m.addVar(name="hours_worked_by_Hank", lb=0, vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(7.22*x1**2 + 4.56*x1*x2 + 9.3*x1 + 7.44*x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(5*x1 + 14*x2 >= 21, name="productivity_rating")
m.addConstr(3*x1 + 4*x2 >= 20, name="paperwork_competence_rating")
m.addConstr(6*x1 - 6*x2 >= 0, name="hours_worked_constraint")
m.addConstr(25*x1**2 + 196*x2**2 <= 114, name="productivity_rating_squared")
m.addConstr(3*x1 + 4*x2 <= 49, name="paperwork_competence_rating_upper_bound")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Bobby: ", x1.varValue)
    print("Hours worked by Hank: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 8: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'hours worked by Bobby'), ('x2', 'hours worked by Hank')],
    'objective_function': '7.22*x1^2 + 4.56*x1*x2 + 9.3*x1 + 7.44*x2',
    'constraints': [
        '5*x1 + 14*x2 >= 21',
        '3*x1 + 4*x2 >= 20',
        '6*x1 - 6*x2 >= 0',
        '25*x1^2 + 196*x2^2 <= 114',
        '3*x1 + 4*x2 <= 49'
    ]
}
```