To find the vertex of the new function $g(x) = f(x + 2) + 2$, let's analyze how this transformation affects the vertex of the original function $f(x)$.

Step-by-Step Solution

1. Original Vertex of $f(x)$:

   • The vertex of $f(x)$ is at $(4, -5)$.

2. Horizontal Shift due to $f(x + 2)$:

   • Replacing $x$ with $x + 2$ in $f(x)$ shifts the graph horizontally to the left by 2 units.
   • Therefore, the $x$-coordinate of the vertex shifts from $4$ to $4 - 2 = 2$.

3. Vertical Shift due to $+2$:

   • Adding 2 to $f(x + 2)$ results in a vertical shift upward by 2 units.
   • Thus, the $y$-coordinate of the vertex shifts from $-5$ to $-5 + 2 = -3$.

4. New Vertex of $g(x)$:

   • After both transformations, the vertex of $g(x)$ is at $(2, -3)$.

Final Answer

The vertex of the function $g(x) = f(x + 2) + 2$ is: $(2, -3)$

Would you like further details on transformations or have any questions?

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Here are some additional questions related to this problem:

1. How does changing $f(x)$ to $f(x - h) + k$ generally affect the graph of a function?
2. What happens to the vertex of $f(x)$ if we change $g(x) = f(x + 2) - 3$?
3. How would a reflection over the x-axis affect the vertex of $f(x)$?
4. What impact does multiplying $f(x)$ by a constant have on the graph’s shape and vertex position?
5. Can we determine the axis of symmetry for $g(x)$ based on the vertex?

Tip: Remember, horizontal shifts in functions affect the x-coordinates, while vertical shifts adjust the y-coordinates of important points like the vertex.