The problem in the image involves transforming a function $f(x) = x^2 - 3x$ by shifting it 4 units to the left and 1 unit upward to obtain a new function $g(x)$.

Step 1: Shift 4 units to the left

Shifting a function to the left by 4 units is done by replacing $x$ with $x + 4$. So, we have: $f(x + 4) = (x + 4)^2 - 3(x + 4)$

Step 2: Expand the expression

Now expand both terms: $(x + 4)^2 = x^2 + 8x + 16$ $-3(x + 4) = -3x - 12$ Thus, $f(x + 4) = x^2 + 8x + 16 - 3x - 12 = x^2 + 5x + 4$

Step 3: Shift 1 unit upward

Shifting the function upward by 1 unit means adding 1 to the entire expression: $g(x) = (x^2 + 5x + 4) + 1 = x^2 + 5x + 5$

Final Answer:

The transformed function $g(x)$ is $x^2 + 5x + 5$, which corresponds to option E.

Would you like more details or have any questions about the steps?

Here are 5 related questions to further explore:

1. How does shifting a function horizontally and vertically affect its graph?
2. What happens to the coefficients when a function is shifted?
3. Can this transformation be applied to any polynomial function?
4. How would the transformation differ for a cubic function?
5. How do you graph a function after applying such transformations?

Tip: When shifting functions, always remember to adjust the variable inside the function for horizontal shifts and the entire expression for vertical shifts.