## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic representation of the problem, including the variables, objective function, and constraints.

## Step 2: Identify the variables and their corresponding symbolic notation
The variables are 'granola bars', 'protein bars', and 'bananas', which we will denote as $x_0$, $x_1$, and $x_2$ respectively.

## Step 3: Translate the objective function into algebraic terms with symbolic variables
The objective function to minimize is $7x_0^2 + 8x_0x_1 + 1x_0x_2 + 5x_1^2 + 8x_1x_2 + 5x_2^2 + 6x_0 + 3x_1 + 5x_2$.

## 4: List the constraints in semi-algebraic form with symbolic variables
The constraints are:
- $4x_0 + 4x_1 + 2x_2 \leq 138$ (grams of fiber)
- $7x_0 + 4x_1 + 6x_2 \leq 51$ (grams of carbohydrates)
- $x_0^2 + x_2^2 \geq 34$ (grams of fiber from granola bars and bananas)
- $x_0^2 + x_1^2 \geq 43$ (grams of fiber from granola bars and protein bars)
- $x_0 + x_1 + x_2 \geq 43$ (grams of fiber from all sources)
- $7x_0 + 4x_1 \geq 14$ (grams of carbohydrates from granola bars and protein bars)
- $4x_1^2 + 6x_2^2 \geq 7$ (grams of carbohydrates from protein bars and bananas)
- $7x_0^2 + 5x_1^2 + 5x_2^2 \geq 15$ (grams of carbohydrates from all sources squared)
- $7x_0 + 4x_1 + 6x_2 \geq 15$ (grams of carbohydrates from all sources)
- $7x_1 - 8x_2 \geq 0$
- $-6x_0 + 6x_1 \geq 0$
- $4x_1 + 6x_2 \leq 18$ (carbohydrates from protein bars and bananas)

## 5: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x0', 'granola bars'), ('x1', 'protein bars'), ('x2', 'bananas')],
    'objective_function': '7*x0^2 + 8*x0*x1 + x0*x2 + 5*x1^2 + 8*x1*x2 + 5*x2^2 + 6*x0 + 3*x1 + 5*x2',
    'constraints': [
        '4*x0 + 4*x1 + 2*x2 <= 138',
        '7*x0 + 4*x1 + 6*x2 <= 51',
        'x0^2 + x2^2 >= 34',
        'x0^2 + x1^2 >= 43',
        'x0 + x1 + x2 >= 43',
        '7*x0 + 4*x1 >= 14',
        '4*x1^2 + 6*x2^2 >= 7',
        '7*x0^2 + 5*x1^2 + 5*x2^2 >= 15',
        '7*x0 + 4*x1 + 6*x2 >= 15',
        '7*x1 - 8*x2 >= 0',
        '-6*x0 + 6*x1 >= 0',
        '4*x1 + 6*x2 <= 18'
    ]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi

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

# Define the variables
x0 = m.addVar(name="granola_bars", lb=0)
x1 = m.addVar(name="protein_bars", lb=0)
x2 = m.addVar(name="bananas", lb=0)

# Set the objective function
m.setObjective(7*x0**2 + 8*x0*x1 + x0*x2 + 5*x1**2 + 8*x1*x2 + 5*x2**2 + 6*x0 + 3*x1 + 5*x2, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(4*x0 + 4*x1 + 2*x2 <= 138)
m.addConstr(7*x0 + 4*x1 + 6*x2 <= 51)
m.addConstr(x0**2 + x2**2 >= 34)
m.addConstr(x0**2 + x1**2 >= 43)
m.addConstr(x0 + x1 + x2 >= 43)
m.addConstr(7*x0 + 4*x1 >= 14)
m.addConstr(4*x1**2 + 6*x2**2 >= 7)
m.addConstr(7*x0**2 + 5*x1**2 + 5*x2**2 >= 15)
m.addConstr(7*x0 + 4*x1 + 6*x2 >= 15)
m.addConstr(7*x1 - 8*x2 >= 0)
m.addConstr(-6*x0 + 6*x1 >= 0)
m.addConstr(4*x1 + 6*x2 <= 18)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Granola bars: ", x0.varValue)
    print("Protein bars: ", x1.varValue)
    print("Bananas: ", x2.varValue)
else:
    print("The model is infeasible")
```