Here's the formulation of the linear program and the Gurobi code to solve it:

**Decision Variables:**

* `x`: Number of shampoo bottles filled
* `y`: Number of conditioner bottles filled

**Objective Function:**

Maximize profit: `5x + 6y`

**Constraints:**

* Shampoo minimum: `x >= 50`
* Conditioner minimum: `y >= 30`
* Time constraint: `3x + 4y <= 300`
* Non-negativity: `x, y >= 0`


```python
import gurobipy as gp
from gurobipy import GRB

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

# Create variables
x = m.addVar(lb=0, vtype=GRB.INTEGER, name="shampoo_bottles") # Integer number of shampoo bottles
y = m.addVar(lb=0, vtype=GRB.INTEGER, name="conditioner_bottles") # Integer number of conditioner bottles


# Set objective
m.setObjective(5*x + 6*y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x >= 50, "shampoo_min")
m.addConstr(y >= 30, "conditioner_min")
m.addConstr(3*x + 4*y <= 300, "time_constraint")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print(f"Optimal Solution Found:")
    print(f"Number of shampoo bottles: {x.x}")
    print(f"Number of conditioner bottles: {y.x}")
    print(f"Maximum Profit: ${m.objVal}")
elif m.status == GRB.INFEASIBLE:
    print("Model is infeasible. No solution exists.")
else:
    print(f"Optimization terminated with status: {m.status}")

```
