To solve the optimization problem described, we need to formulate it as a linear programming problem and then use Gurobi to find the optimal solution. The objective is to maximize the function $8x_0 + 7x_1 + 3x_2$, where $x_0$ represents the number of chili plants, $x_1$ represents the number of pansies, and $x_2$ represents the number of roses.

The constraints given can be summarized as follows:
- Growth speed constraints:
  - $5x_0 + 2x_2 \leq 13$
  - $4x_1 + 2x_2 \leq 11$
  - $5x_0 + 4x_1 + 2x_2 \leq 26$
- Cost constraints:
  - $5x_0 + x_1 \leq 21$
  - $5x_0 + 2x_2 \leq 11$
  - $x_1 + 2x_2 \leq 23$
  - $5x_0 + x_1 + 2x_2 \leq 28$

All variables ($x_0, x_1, x_2$) must be non-negative integers.

Here is the Gurobi code to solve this problem:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Create variables
x0 = m.addVar(vtype=GRB.INTEGER, name="chili_plants")
x1 = m.addVar(vtype=GRB.INTEGER, name="pansies")
x2 = m.addVar(vtype=GRB.INTEGER, name="roses")

# Set the objective function
m.setObjective(8*x0 + 7*x1 + 3*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(5*x0 + 2*x2 <= 13, "growth_speed_1")
m.addConstr(4*x1 + 2*x2 <= 11, "growth_speed_2")
m.addConstr(5*x0 + 4*x1 + 2*x2 <= 26, "total_growth_speed")
m.addConstr(5*x0 + x1 <= 21, "cost_1")
m.addConstr(5*x0 + 2*x2 <= 11, "cost_2")
m.addConstr(x1 + 2*x2 <= 23, "cost_3")
m.addConstr(5*x0 + x1 + 2*x2 <= 28, "total_cost")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Chili plants: {x0.x}")
    print(f"Pansies: {x1.x}")
    print(f"Roses: {x2.x}")
    print(f"Objective value: {m.objVal}")
else:
    print("No optimal solution found.")
```