## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Dale' and 'hours worked by Hank', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $4x_1^2 + 6x_1x_2 + 7x_1$.

## 3: Define the constraints in symbolic notation
The constraints given are:
- $4x_1 + 4x_2 \geq 29$
- $-8x_1 + 6x_2 \geq 0$
- $4x_1 + 4x_2 \leq 70$

## 4: Consider the computer competence rating constraints
The computer competence ratings for Dale and Hank are both 4. The total combined rating constraint is already considered in the constraints above.

## 5: Determine the symbolic representation of the problem
The symbolic representation is:
- Variables: $x_1$ for 'hours worked by Dale', $x_2$ for 'hours worked by Hank'
- Objective function: $4x_1^2 + 6x_1x_2 + 7x_1$
- Constraints:
  1. $4x_1 + 4x_2 \geq 29$
  2. $-8x_1 + 6x_2 \geq 0$
  3. $4x_1 + 4x_2 \leq 70$

## 6: Create the Gurobi model and variables
We will use Gurobi to solve this optimization problem.

## 7: Implement the Gurobi code
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="hours_worked_by_Dale", lb=-gp.GRB.INFINITY)  # No lower bound specified
x2 = m.addVar(name="hours_worked_by_Hank", lb=-gp.GRB.INFINITY)  # No lower bound specified

# Define the objective function
m.setObjective(4*x1**2 + 6*x1*x2 + 7*x1, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(4*x1 + 4*x2 >= 29, name="min_computer_competence_rating")
m.addConstr(-8*x1 + 6*x2 >= 0, name="linear_constraint")
m.addConstr(4*x1 + 4*x2 <= 70, name="max_computer_competence_rating")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Dale: {x1.varValue}")
    print(f"Hours worked by Hank: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 8: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x1', 'hours worked by Dale'), ('x2', 'hours worked by Hank')],
    'objective_function': '4*x1^2 + 6*x1*x2 + 7*x1',
    'constraints': [
        '4*x1 + 4*x2 >= 29',
        '-8*x1 + 6*x2 >= 0',
        '4*x1 + 4*x2 <= 70'
    ]
}
```