## Step 1: Define the optimization problem
The problem is to maximize the objective function: $1 \times \text{fruit salads}^2 + 3 \times \text{fruit salads} \times \text{strawberries} + 5 \times \text{fruit salads} \times \text{peanutbutter sandwiches} + 9 \times \text{strawberries}^2 + 8 \times \text{fruit salads} + 8 \times \text{strawberries}$, subject to various constraints on umami index, tastiness rating, cost, and calcium content.

## Step 2: Identify the variables and their attributes
The variables are:
- $\text{fruit salads}$ with attributes: umami index $= 6$, tastiness rating $= 8$, cost $= 5$, calcium $= 15$
- $\text{strawberries}$ with attributes: umami index $= 15$, tastiness rating $= 5$, cost $= 1$, calcium $= 7$
- $\text{peanutbutter sandwiches}$ with attributes: umami index $= 5$, tastiness rating $= 20$, cost $= 19$, calcium $= 12$

## 3: Formulate the objective function
The objective function to maximize is: $1 \times \text{fruit salads}^2 + 3 \times \text{fruit salads} \times \text{strawberries} + 5 \times \text{fruit salads} \times \text{peanutbutter sandwiches} + 9 \times \text{strawberries}^2 + 8 \times \text{fruit salads} + 8 \times \text{strawberries}$

## 4: Define the constraints
Constraints include:
- Umami index constraints: $6 \times \text{fruit salads} + 15 \times \text{strawberries} \geq 68$, $15 \times \text{strawberries} + 5 \times \text{peanutbutter sandwiches} \geq 37$, $6 \times \text{fruit salads} + 5 \times \text{peanutbutter sandwiches} \geq 27$, $15 \times \text{strawberries} + 5 \times \text{peanutbutter sandwiches} \leq 196$
- Tastiness rating constraints: $8 \times \text{fruit salads} + 20 \times \text{peanutbutter sandwiches} \leq 140$, $8 \times \text{fruit salads} + 5 \times \text{strawberries} \leq 122$, $8 \times \text{fruit salads} + 5 \times \text{strawberries} + 20 \times \text{peanutbutter sandwiches} \leq 122$
- Cost constraints: $5 \times \text{fruit salads} + 19 \times \text{peanutbutter sandwiches} \geq 78$, $5 \times \text{fruit salads} + 1 \times \text{strawberries} \geq 40$, $5 \times \text{fruit salads} + 1 \times \text{strawberries} + 19 \times \text{peanutbutter sandwiches} \leq 232$, $5 \times \text{fruit salads}^2 + 1 \times \text{strawberries}^2 \leq 232$
- Calcium constraints: $15 \times \text{fruit salads}^2 + 7 \times \text{strawberries}^2 \geq 49$, $7 \times \text{strawberries} + 12 \times \text{peanutbutter sandwiches} \geq 88$, $15 \times \text{fruit salads} + 12 \times \text{peanutbutter sandwiches} \geq 138$, $7 \times \text{strawberries}^2 + 12 \times \text{peanutbutter sandwiches}^2 \leq 266$, $15 \times \text{fruit salads}^2 + 7 \times \text{strawberries}^2 \leq 160$, $15 \times \text{fruit salads} + 7 \times \text{strawberries} + 12 \times \text{peanutbutter sandwiches} \leq 160$
- Variable constraints: $\text{fruit salads}, \text{strawberries}, \text{peanutbutter sandwiches}$ are integers.

## 5: Implement the problem in Gurobi
```python
import gurobipy as gp

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

# Define the variables
fruit_salads = m.addVar(name="fruit_salads", vtype=gp.GRB.INTEGER)
strawberries = m.addVar(name="strawberries", vtype=gp.GRB.INTEGER)
peanutbutter_sandwiches = m.addVar(name="peanutbutter_sandwiches", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(fruit_salads**2 + 3 * fruit_salads * strawberries + 5 * fruit_salads * peanutbutter_sandwiches + 
               9 * strawberries**2 + 8 * fruit_salads + 8 * strawberries, gp.GRB.MAXIMIZE)

# Umami index constraints
m.addConstr(6 * fruit_salads + 15 * strawberries >= 68)
m.addConstr(15 * strawberries + 5 * peanutbutter_sandwiches >= 37)
m.addConstr(6 * fruit_salads + 5 * peanutbutter_sandwiches >= 27)
m.addConstr(15 * strawberries + 5 * peanutbutter_sandwiches <= 196)

# Tastiness rating constraints
m.addConstr(8 * fruit_salads + 20 * peanutbutter_sandwiches <= 140)
m.addConstr(8 * fruit_salads + 5 * strawberries <= 122)
m.addConstr(8 * fruit_salads + 5 * strawberries + 20 * peanutbutter_sandwiches <= 122)

# Cost constraints
m.addConstr(5 * fruit_salads + 19 * peanutbutter_sandwiches >= 78)
m.addConstr(5 * fruit_salads + strawberries >= 40)
m.addConstr(5 * fruit_salads + strawberries + 19 * peanutbutter_sandwiches <= 232)
m.addConstr(5 * fruit_salads**2 + strawberries**2 <= 232)

# Calcium constraints
m.addConstr(15 * fruit_salads**2 + 7 * strawberries**2 >= 49)
m.addConstr(7 * strawberries + 12 * peanutbutter_sandwiches >= 88)
m.addConstr(15 * fruit_salads + 12 * peanutbutter_sandwiches >= 138)
m.addConstr(7 * strawberries**2 + 12 * peanutbutter_sandwiches**2 <= 266)
m.addConstr(15 * fruit_salads**2 + 7 * strawberries**2 <= 160)
m.addConstr(15 * fruit_salads + 7 * strawberries + 12 * peanutbutter_sandwiches <= 160)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Fruit Salads: ", fruit_salads.varValue)
    print("Strawberries: ", strawberries.varValue)
    print("Peanutbutter Sandwiches: ", peanutbutter_sandwiches.varValue)
else:
    print("The model is infeasible")
```