## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. The variables are 'cafeteria chairs', 'red highlighters', 'mechanical pencils', and '3D printers', which we can denote as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Translate the objective function into algebraic terms
The objective function to maximize is: $7x_0^2 + 6x_0x_1 + 2x_0x_2 + 4x_0x_3 + 2x_1^2 + 4x_1x_2 + x_1x_3 + 3x_2^2 + x_2x_3 + 8x_3^2 + 9x_0 + 3x_1 + x_2 + 5x_3$.

## 3: List the constraints in algebraic terms
The constraints are:
- $13x_0 + 4x_1 + 11x_2 + 10x_3 \leq 100$ (storage space)
- $10x_0 + 9x_1 + 12x_2 + 2x_3 \leq 113$ (workplace safety impact)
- $x_0 + x_1 \geq 15$ (storage space)
- $x_0 + x_3 \geq 12$ (storage space)
- $x_2 + x_3 \geq 18$ (storage space)
- $x_1 + x_2 + x_3 \geq 15$ (storage space)
- $x_0 + x_1 + x_3 \geq 15$ (storage space)
- $x_0 + x_1 + x_2 \geq 15$ (storage space)
- $x_0^2 + x_2^2 + x_3^2 \geq 15$ (storage space)
- $x_1 + x_2 + x_3 \geq 22$ (storage space)
- $x_0 + x_1 + x_3 \geq 22$ (storage space)
- $x_0 + x_1 + x_2 \geq 22$ (storage space)
- $x_0^2 + x_2^2 + x_3^2 \geq 22$ (storage space)
- $x_1^2 + x_2^2 + x_3^2 \geq 12$ (storage space)
- $x_0 + x_1 + x_3 \geq 12$ (storage space)
- $x_0 + x_1 + x_2 \geq 12$ (storage space)
- $x_0^2 + x_2^2 + x_3^2 \geq 12$ (storage space)
- $x_1 + x_2 + x_3 \geq 16$ (storage space)
- $x_0 + x_1 + x_3 \geq 16$ (storage space)
- $x_0 + x_1 + x_2 \geq 16$ (storage space)
- $x_0 + x_2 + x_3 \geq 16$ (storage space)
- $x_0^2 + x_3^2 \geq 20$ (workplace safety impact)
- $x_0 + x_1 + x_3 \geq 25$ (workplace safety impact)
- $x_0 + x_2 + x_3 \geq 25$ (workplace safety impact)
- $x_1^2 + x_2^2 + x_3^2 \geq 25$ (workplace safety impact)
- $x_0^2 + x_1^2 + x_3^2 \geq 15$ (workplace safety impact)
- $x_0^2 + x_2^2 + x_3^2 \geq 15$ (workplace safety impact)
- $x_1^2 + x_2^2 + x_3^2 \geq 15$ (workplace safety impact)
- $x_0 + x_1 + x_3 \geq 14$ (workplace safety impact)
- $x_0 + x_2 + x_3 \geq 14$ (workplace safety impact)
- $x_1 + x_2 + x_3 \geq 14$ (workplace safety impact)
- $x_1^2 + x_2^2 \leq 74$ (storage space)
- $x_1^2 + x_3^2 \leq 92$ (storage space)
- $x_0 + x_1 + x_2 + x_3 \leq 92$ (storage space)
- $x_1 + x_3 \leq 30$ (workplace safety impact)
- $x_0 + x_1 \leq 30$ (workplace safety impact)
- $x_0 + x_1 + x_3 \leq 102$ (workplace safety impact)
- $x_0^2 + x_1^2 + x_2^2 \leq 48$ (workplace safety impact)
- $x_1^2 + x_2^2 + x_3^2 \leq 61$ (workplace safety impact)
- $x_0 + x_1 + x_2 + x_3 \leq 61$ (workplace safety impact)
- $x_0, x_1, x_2, x_3$ are integers.

## 4: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'cafeteria chairs'), 
        ('x1', 'red highlighters'), 
        ('x2', 'mechanical pencils'), 
        ('x3', '3D printers')
    ], 
    'objective_function': '7*x0^2 + 6*x0*x1 + 2*x0*x2 + 4*x0*x3 + 2*x1^2 + 4*x1*x2 + x1*x3 + 3*x2^2 + x2*x3 + 8*x3^2 + 9*x0 + 3*x1 + x2 + 5*x3', 
    'constraints': [
        '13*x0 + 4*x1 + 11*x2 + 10*x3 <= 100',
        '10*x0 + 9*x1 + 12*x2 + 2*x3 <= 113',
        'x0 + x1 >= 15',
        'x0 + x3 >= 12',
        'x2 + x3 >= 18',
        'x1 + x2 + x3 >= 15',
        'x0 + x1 + x3 >= 15',
        'x0 + x1 + x2 >= 15',
        'x0^2 + x2^2 + x3^2 >= 15',
        'x1 + x2 + x3 >= 22',
        'x0 + x1 + x3 >= 22',
        'x0 + x1 + x2 >= 22',
        'x0^2 + x2^2 + x3^2 >= 22',
        'x1^2 + x2^2 + x3^2 >= 12',
        'x0 + x1 + x3 >= 12',
        'x0 + x1 + x2 >= 12',
        'x0^2 + x2^2 + x3^2 >= 12',
        'x1 + x2 + x3 >= 16',
        'x0 + x1 + x3 >= 16',
        'x0 + x1 + x2 >= 16',
        'x0 + x2 + x3 >= 16',
        'x0^2 + x3^2 >= 20',
        'x0 + x1 + x3 >= 25',
        'x0 + x2 + x3 >= 25',
        'x1^2 + x2^2 + x3^2 >= 25',
        'x0^2 + x1^2 + x3^2 >= 15',
        'x0^2 + x2^2 + x3^2 >= 15',
        'x1^2 + x2^2 + x3^2 >= 15',
        'x0 + x1 + x3 >= 14',
        'x0 + x2 + x3 >= 14',
        'x1 + x2 + x3 >= 14',
        'x1^2 + x2^2 <= 74',
        'x1^2 + x3^2 <= 92',
        'x0 + x1 + x2 + x3 <= 92',
        'x1 + x3 <= 30',
        'x0 + x1 <= 30',
        'x0 + x1 + x3 <= 102',
        'x0^2 + x1^2 + x2^2 <= 48',
        'x1^2 + x2^2 + x3^2 <= 61',
        'x0 + x1 + x2 + x3 <= 61'
    ]
}
```

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

def optimize_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="cafeteria_chairs", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="red_highlighters", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="mechanical_pencils", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="3D_printers", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(7*x0**2 + 6*x0*x1 + 2*x0*x2 + 4*x0*x3 + 2*x1**2 + 4*x1*x2 + x1*x3 + 3*x2**2 + x2*x3 + 8*x3**2 + 9*x0 + 3*x1 + x2 + 5*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(13*x0 + 4*x1 + 11*x2 + 10*x3 <= 100)
    model.addConstr(10*x0 + 9*x1 + 12*x2 + 2*x3 <= 113)
    model.addConstr(x0 + x1 >= 15)
    model.addConstr(x0 + x3 >= 12)
    model.addConstr(x2 + x3 >= 18)
    model.addConstr(x1 + x2 + x3 >= 15)
    model.addConstr(x0 + x1 + x3 >= 15)
    model.addConstr(x0 + x1 + x2 >= 15)
    model.addConstr(x0**2 + x2**2 + x3**2 >= 15)
    model.addConstr(x1 + x2 + x3 >= 22)
    model.addConstr(x0 + x1 + x3 >= 22)
    model.addConstr(x0 + x1 + x2 >= 22)
    model.addConstr(x0**2 + x2**2 + x3**2 >= 22)
    model.addConstr(x1**2 + x2**2 + x3**2 >= 12)
    model.addConstr(x0 + x1 + x3 >= 12)
    model.addConstr(x0 + x1 + x2 >= 12)
    model.addConstr(x0**2 + x2**2 + x3**2 >= 12)
    model.addConstr(x1 + x2 + x3 >= 16)
    model.addConstr(x0 + x1 + x3 >= 16)
    model.addConstr(x0 + x1 + x2 >= 16)
    model.addConstr(x0 + x2 + x3 >= 16)
    model.addConstr(x0**2 + x3**2 >= 20)
    model.addConstr(x0 + x1 + x3 >= 25)
    model.addConstr(x0 + x2 + x3 >= 25)
    model.addConstr(x1**2 + x2**2 + x3**2 >= 25)
    model.addConstr(x0**2 + x1**2 + x3**2 >= 15)
    model.addConstr(x0**2 + x2**2 + x3**2 >= 15)
    model.addConstr(x1**2 + x2**2 + x3**2 >= 15)
    model.addConstr(x0 + x1 + x3 >= 14)
    model.addConstr(x0 + x2 + x3 >= 14)
    model.addConstr(x1 + x2 + x3 >= 14)
    model.addConstr(x1**2 + x2**2 <= 74)
    model.addConstr(x1**2 + x3**2 <= 92)
    model.addConstr(x0 + x1 + x2 + x3 <= 92)
    model.addConstr(x1 + x3 <= 30)
    model.addConstr(x0 + x1 <= 30)
    model.addConstr(x0 + x1 + x3 <= 102)
    model.addConstr(x0**2 + x1**2 + x2**2 <= 48)
    model.addConstr(x1**2 + x2**2 + x3**2 <= 61)
    model.addConstr(x0 + x1 + x2 + x3 <= 61)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Cafeteria chairs: ", x0.varValue)
        print("Red highlighters: ", x1.varValue)
        print("Mechanical pencils: ", x2.varValue)
        print("3D printers: ", x3.varValue)
    else:
        print("The model is infeasible")

optimize_problem()
```