To identify where Hyung-eun first makes an error, let's review her steps carefully:
The original equation is:
\[ x^2 - 4x - 12 = 0 \]
Step 1:
\[ x^2 + (-6x) + 2x - 12 = 0 \]
This step is incorrect. The terms \(-6x\) and \(2x\) do not properly combine to give the middle term \(-4x\). The correct grouping should have simplified to:
\[ x^2 - 6x + 2x - 12 = 0 \]
which can be rearranged correctly to combine to \(x^2 - 4x - 12 = 0\).
Step 2:
In her step, she attempts to factor the expression. This is based on her incorrect step 1.
Step 3:
She factors the incorrect grouping, which is also wrong since it stems from the error in step 1.
Step 4:
Here, she writes out the factors based on incorrect grouping, also stemming from errors in the prior steps.
The first clear error occurs in Step 1 because she incorrectly rearranged the equation's quadratic coefficients, leading to all subsequent errors. Therefore, the correct answer is:
Step 1.