We are given several transformations to apply to the function $f(x) = x^2$. Let's apply them step by step in the following order:

1. Shift left 2 units: Shifting a function left by 2 units means replacing $x$ with $x + 2$. So, the function becomes: $f(x) = (x + 2)^2$

2. Reflect across the y-axis: Reflecting across the y-axis means replacing $x$ with $-x$. Now, the function becomes: $f(x) = (-x + 2)^2 = (2 - x)^2$

3. Shift down 5 units: Shifting down means subtracting 5 from the function. Thus, the function becomes: $f(x) = (2 - x)^2 - 5$

4. Vertical scaling by a factor of 4: Scaling vertically means multiplying the entire function by 4. This results in: $f(x) = 4((2 - x)^2 - 5) = 4(2 - x)^2 - 20$

5. Reflect across the x-axis: Reflecting across the x-axis means multiplying the entire function by $-1$. The final function is: $f(x) = -4(2 - x)^2 + 20$

Thus, the transformed function is:

$f(x) = -4(2 - x)^2 + 20$

Would you like further explanation or have any questions?

Here are 5 related questions to enhance understanding:

1. How does reflecting a function across the x-axis affect its graph?
2. What is the general form of a parabola after a horizontal shift and reflection?
3. How would you shift a function right by 3 units instead of left?
4. What does vertical stretching or compressing do to a quadratic function?
5. How can we recognize the transformations of a function from its equation alone?

Tip: Always apply horizontal transformations (shifts and reflections) before vertical transformations (scaling, shifts, or reflections) to avoid confusion.