If \(\mathrm{k(m^2 - 2m) - 3k(m^2 - 2m) = -12}\), and m^2 - 2m = 4, what is the value...

1. INFER the key strategy from the problem structure

• Given information:
  • \(\mathrm{k(m^2 - 2m) - 3k(m^2 - 2m) = -12}\)
  • \(\mathrm{m^2 - 2m = 4}\)

• Key insight: Both terms on the left contain the same expression \(\mathrm{(m^2 - 2m)}\), so we can factor it out rather than expanding everything

2. SIMPLIFY by factoring out the common term

• Factor out \(\mathrm{(m^2 - 2m)}\):

\(\mathrm{k(m^2 - 2m) - 3k(m^2 - 2m) = (m^2 - 2m)(k - 3k)}\)

• Combine the k terms:

\(\mathrm{(m^2 - 2m)(-2k) = -12}\)

3. INFER that substitution is the next step

• Since we know \(\mathrm{m^2 - 2m = 4}\), substitute this value:

\(\mathrm{4(-2k) = -12}\)

4. SIMPLIFY to solve for k

• Multiply: \(\mathrm{-8k = -12}\)
• Divide both sides by -8: \(\mathrm{k = \frac{-12}{-8} = \frac{3}{2}}\)

Answer: D (3/2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the factoring opportunity with \(\mathrm{(m^2 - 2m)}\) as a common term

Students often try to expand \(\mathrm{k(m^2 - 2m)}\) and \(\mathrm{3k(m^2 - 2m)}\) separately, leading to \(\mathrm{km^2 - 2km - 3km^2 + 6km = -12}\), which simplifies to \(\mathrm{-2km^2 + 4km = -12}\). From here, they get stuck because they can't use the given condition \(\mathrm{m^2 - 2m = 4}\) directly, and the algebra becomes unnecessarily complicated.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when combining terms

Students correctly factor out \(\mathrm{(m^2 - 2m)}\) but then incorrectly calculate \(\mathrm{k - 3k}\). Some get \(\mathrm{k - 3k = -2k}\) (correct), but others might get \(\mathrm{k - 3k = 2k}\) or \(\mathrm{4k}\). When this happens with the incorrect sign, they might get \(\mathrm{8k = -12}\) instead of \(\mathrm{-8k = -12}\), leading to \(\mathrm{k = -\frac{3}{2}}\).

This may lead them to select Choice A (-3/2).

The Bottom Line:

This problem tests whether students can recognize patterns and use substitution efficiently. The key insight is seeing \(\mathrm{(m^2 - 2m)}\) as a single "chunk" that can be factored out, rather than expanding everything into individual terms.