Hyung-eun is solving the quadratic equation x^2−4x−12=0 by grouping. Her work is shown below. At which step does Hyung-eun first make an error?

Hyung-eun's first error occurs in Step 4.

Here's a corrected breakdown of her work:

1. Starting with the quadratic equation \(x^2 - 4x - 12 = 0\).
2. In Step 1, Hyung-eun tries to rearrange the equation: \(x^2 + (-6x) + 2x - 12 = 0\).
   • This step is okay, but the grouping isn't correct for this quadratic equation.
3. In Step 2, she groups terms: \((x^2 + (-6x)) + (2x - 12) = 0\).
   • This grouping is not ideal but mathematically still follows from Step 1.
4. In Step 3, she factored out \(x\) from the first group and \(2\) from the second group: \(x(x - 6) + 2(x - 6) = 0\).
   • This step is actually valid, as both groups share a common factor.

However, the error occurs in Step 4. She incorrectly factors it as \((x + 6)(x - 2) = 0\), which is incorrect. The correct factor should have been \( (x - 6 + 2)(x - 6)\) leading to \( (x - 6)(x + 2) = 0\).

Hence the correct factorization should yield \( (x - 6)(x + 2) = 0\), which gives the solutions as \( x = 6 \) or \( x = -2 \).

Thus, the first error is in Step 4 where she factors incorrectly.