The inequality provided is:

$y < 2(x + 2)^2 + 3$

Step-by-Step Solution:

1. Identify the equation of the parabola: The equation $y = 2(x + 2)^2 + 3$ represents a parabola. It opens upwards because the coefficient of $(x + 2)^2$ is positive.

   • Vertex: The vertex of this parabola is at $(-2, 3)$, since the equation is in vertex form $y = a(x - h)^2 + k$, where $h = -2$ and $k = 3$.
   • Direction and Stretch: The parabola opens upward because of the positive leading coefficient $2$, and the factor of $2$ makes it narrower than a standard parabola.

2. Graphing the boundary: The boundary is the equation $y = 2(x + 2)^2 + 3$, but because the inequality is strict (i.e., $y <$), we will use a dashed line to represent the boundary, indicating that points on the line are not included in the solution.

3. Determine the shaded region: Since the inequality is $y < 2(x + 2)^2 + 3$, you need to shade the region below the parabola. This represents all points where $y$ is less than the value of the parabola at any given $x$.

Graphing Steps:

1. Draw the dashed parabola:

   • The vertex is at $(-2, 3)$.
   • Plot a few points for accuracy. For example:
     • At $x = -2$, $y = 3$ (the vertex).
     • At $x = -1$, $y = 5$.
     • At $x = 0$, $y = 11$.
     • At $x = -3$, $y = 5$.
     • At $x = -4$, $y = 11$.

2. Shade below the parabola: Since the inequality is $y <$, shade the area below the parabola.

Let me know if you'd like more details on plotting or any specific parts of the solution!

Here are five related questions to deepen your understanding:

1. What happens if the inequality was $y \leq 2(x + 2)^2 + 3$?
2. How would the graph change if the coefficient of $(x + 2)^2$ were negative?
3. How do you determine which region to shade in inequalities?
4. What if the vertex of the parabola were at $(0, 0)$?
5. Can you explain the effect of the 3 in $y = 2(x + 2)^2 + 3$ on the graph?

Tip: Always plot the vertex and a few additional points when graphing parabolas to ensure accuracy!