To solve this problem, we need to analyze the given inequalities and determine their graphical representations. Here's the step-by-step process:

System of Inequalities:

1. First inequality: $x^2 + y < -5$

   • This represents a parabola $y = -x^2 - 5$, but since it's a strict inequality ($<$), the region below this curve (not including the boundary) is shaded.

2. Second inequality: $y > 5x + 7$

   • This is a linear inequality with slope $5$ and $y$-intercept $7$. The shaded region lies above the line $y = 5x + 7$.

Steps to Graph:

• Plot the parabola $y = -x^2 - 5$, and shade the region below it.
• Plot the line $y = 5x + 7$, and shade the region above it.
• The solution region is the overlap of the shaded areas from the two inequalities.

Intersection Points:

To find points of intersection, solve the equations: $y = -x^2 - 5$ $y = 5x + 7$

Equating: $-x^2 - 5 = 5x + 7$ $-x^2 - 5x - 12 = 0$ $x^2 + 5x + 12 = 0$

The discriminant ($\Delta$) of this quadratic is: $\Delta = b^2 - 4ac = 5^2 - 4(1)(12) = 25 - 48 = -23$

Since the discriminant is negative, there are no real solutions. Therefore, the parabola and line do not intersect in the real plane.

Graph Characteristics:

• The parabola opens downward, and its vertex is at $(0, -5)$.
• The line has a steep positive slope with a $y$-intercept at $7$.

Thus, the correct graph will show:

1. The shaded region under the parabola.
2. The shaded region above the line.
3. No overlap or intersection between the line and parabola in the real plane.

Would you like a graphical representation or further clarification? Here are related questions for further practice:

1. How can you verify inequalities by substituting test points?
2. What happens if the parabola $y = -x^2 - 5$ is shifted vertically?
3. How does the slope of the line $y = 5x + 7$ affect the shaded region?
4. What if the inequality signs in the system were reversed?
5. How do imaginary solutions in quadratic equations manifest in graphing?

Tip: Always check for boundary conditions to ensure the solution region is correctly identified.