The problem is asking for the transformations required to graph $g(x) = -2(x + 3)^2 + 1$ using the base function $f(x) = x^2$.

Let's break down the transformations:

1. Shift the graph horizontally to the left by 3 units:

   • The term $(x + 3)$ inside the square indicates a horizontal shift 3 units to the left.

2. Vertical reflection:

   • The negative sign in front of the 2, $-2(x + 3)^2$, reflects the graph about the x-axis.

3. Vertical stretch by a factor of 2:

   • The factor of 2 in front of $(x + 3)^2$ stretches the graph vertically by a factor of 2.

4. Shift the graph vertically upward by 1 unit:

   • The constant $+1$ at the end shifts the graph up by 1 unit.

Given this breakdown, the correct transformations are:

• A: Shift the graph 3 units to the left.
• B: Shift the graph 1 unit up (this option is correct but labeled incorrectly in the image; "down" should be "up").
• E: Reflect the graph about the x-axis.
• F: Stretch the graph vertically by a factor of 2.

Let me know if you'd like more details or explanations!

Here are 5 related questions for deeper understanding:

1. What happens if the graph is shifted 3 units to the right instead of left?
2. How does a horizontal stretch affect the graph of a quadratic function?
3. What is the effect of a vertical compression on the graph of $f(x) = x^2$?
4. How would the graph change if the constant was $-1$ instead of $+1$?
5. How does a horizontal reflection differ from a vertical reflection?

Tip: Always check the signs inside and outside the parentheses to determine whether a graph shifts left or right, and whether it stretches or compresses.