The question asks to find the equation of the function $f(x) = x^2$ after it has been vertically stretched by a factor of 3, shifted 5 units to the right, and 4 units up.

Let's break down the transformations:

1. Vertical Stretch by a factor of 3: This will multiply the function by 3, turning $f(x) = x^2$ into $g(x) = 3x^2$.

2. Shift 5 units to the right: To shift the function right, we replace $x$ with $(x - 5)$. So the function becomes $g(x) = 3(x - 5)^2$.

3. Shift 4 units up: Finally, adding 4 units to the function shifts it upward. The equation becomes $g(x) = 3(x - 5)^2 + 4$.

So, the correct answer is: $g(x) = 3(x - 5)^2 + 4$

In the provided choices, this matches the second option: $g(x) = 3(x - 5)^2 + 4$.

Would you like further clarification or details on any of the transformations?

Here are five related questions for further practice:

1. How would the equation change if the function were shifted 3 units left instead of right?
2. What happens to the graph if we vertically compress instead of stretch by a factor of 3?
3. How would the equation look if the function was reflected across the x-axis?
4. How does a horizontal shift affect the vertex of a parabola?
5. What is the effect of adding a constant inside the parentheses, instead of outside?

Tip: When transforming functions, keep track of each step by applying one transformation at a time—this makes it easier to avoid errors.