To solve the given problem, let us determine the inequality represented by the graph.

Step 1: Analyze the graph

• The graph represents a parabola opening upwards.
• The shaded region is inside the parabola, which suggests that the inequality involves ≤ or ≥, depending on the orientation.

Step 2: Find the quadratic equation

The general form of a quadratic function is: $y = ax^2 + bx + c$ From the graph:

• The vertex of the parabola is at (-3, 5), so the vertex form of the parabola is: $y - 5 = a(x + 3)^2$
• From the equation $y - 5 = x^2 + 6x + 9$, the simplified quadratic becomes: $y - 5 = x^2 + 6x$

Step 3: Select the inequality

• The graph shows shading below or inside the parabola. Hence, the inequality should be: $y - 5 \leq x^2 + 6x$

Final Answer:

The correct choice is: $\boxed{y - 5 \leq x^2 + 6x}$

Do you want further clarification or details? Here are some related questions to deepen understanding:

1. What is the vertex form of a parabola, and how is it related to the standard form?
2. How do we determine the orientation of the parabola based on its inequality?
3. What are some key steps for solving quadratic inequality problems?
4. Can you explain the relationship between shading and the inequality symbol?
5. How would the graph change if the inequality were $y - 5 \geq x^2 + 6x$?

Tip: Always verify the orientation and shading of the graph to choose the correct inequality.