To shift the graph of $f(x) = 2x^2$ left by 2 units and down by 4 units, you apply the following transformations:

1. Left 2 units: Replace $x$ with $x + 2$ in the function $f(x)$. This modifies the equation to: $f(x+2) = 2(x+2)^2$

2. Down 4 units: Subtract 4 from the function. This modifies the equation to: $g(x) = 2(x+2)^2 - 4$

Thus, the equation of $g(x)$ is: $g(x) = 2(x+2)^2 - 4$

Would you like me to expand this function, graph it, or provide more explanations?

Here are 5 related questions for further exploration:

1. How would the equation change if the graph were shifted to the right by 3 units and up 5 units?
2. What are the vertex and axis of symmetry for $g(x)$?
3. How do horizontal and vertical shifts affect the domain and range of a quadratic function?
4. Can you derive the transformation rules for other types of functions, like $f(x) = \sqrt{x}$?
5. How does the graph of $g(x) = -2(x+2)^2 - 4$ differ from $g(x) = 2(x+2)^2 - 4$?

Tip: When shifting a graph, always consider the transformations sequentially (horizontal, vertical, reflections, and stretching/compressing).