The given function $f(x) = x^2 - 2x$ undergoes a series of transformations to form $g(x)$. Let's break down each transformation step:

1. Translation 2 units left: This modifies the $x$-term in the function by replacing $x$ with $(x + 2)$. The new function after this step becomes: $f(x+2) = (x+2)^2 - 2(x+2)$ Expanding this: $(x+2)^2 - 2(x+2) = x^2 + 4x + 4 - 2x - 4 = x^2 + 2x$

2. Translation 3 units up: To translate the graph up by 3 units, we add 3 to the function: $f(x+2) + 3 = (x^2 + 2x) + 3 = x^2 + 2x + 3$

3. Reflection in the $y$-axis: Reflecting the graph in the $y$-axis involves replacing $x$ with $-x$: $g(x) = (-x)^2 + 2(-x) + 3 = x^2 - 2x + 3$

Thus, the rule for $g(x)$ is: $g(x) = x^2 - 2x + 3$

Would you like further details or explanations on any step?

Here are 5 follow-up questions you may consider:

1. How does reflecting a graph in the $y$-axis affect its symmetry?
2. What is the effect of translating a graph horizontally by a certain number of units?
3. Can you describe how vertical translations affect the range of a function?
4. How would the transformations change if we reflected the graph in the $x$-axis instead?
5. How do you expand binomials like $(x + 2)^2$ algebraically?

Tip: When performing multiple transformations, always follow the order of operations: apply horizontal transformations, then reflections, and finally vertical translations.