Which expression is equivalent to \(-1(3\mathrm{x}^2 - \mathrm{x} + 5) + (4\mathrm{x}^2 - 2\mathrm{x} - 8)\)?-{x^2 - 3x + 13}x^2...

1. SIMPLIFY by applying the distributive property

• First, distribute -1 to every term in the first polynomial:
  \(-1(3\mathrm{x}^2 - \mathrm{x} + 5) = -3\mathrm{x}^2 + \mathrm{x} - 5\)

• Keep the second polynomial as is: \((4\mathrm{x}^2 - 2\mathrm{x} - 8)\)

2. VISUALIZE by organizing like terms

• Write out the complete expression:
  \(-3\mathrm{x}^2 + \mathrm{x} - 5 + 4\mathrm{x}^2 - 2\mathrm{x} - 8\)

• Group terms by degree and variable:
  \((-3\mathrm{x}^2 + 4\mathrm{x}^2) + (\mathrm{x} - 2\mathrm{x}) + (-5 - 8)\)

3. SIMPLIFY by combining like terms

• x² terms: \(-3\mathrm{x}^2 + 4\mathrm{x}^2 = 1\mathrm{x}^2 = \mathrm{x}^2\)
• x terms: \(\mathrm{x} - 2\mathrm{x} = -1\mathrm{x} = -\mathrm{x}\)
• Constants: \(-5 - 8 = -13\)

• Final result: \(\mathrm{x}^2 - \mathrm{x} - 13\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when distributing the negative coefficient

Students often make mistakes like:

• \(-1(3\mathrm{x}^2 - \mathrm{x} + 5) = -3\mathrm{x}^2 + \mathrm{x} + 5\) (forgetting to make the +5 negative)
• \(-1(3\mathrm{x}^2 - \mathrm{x} + 5) = -3\mathrm{x}^2 - \mathrm{x} - 5\) (making the -x term negative when it should become positive)

These sign errors cascade through the rest of the problem, potentially leading them to select Choice A (\(-\mathrm{x}^2 - 3\mathrm{x} + 13\)) or getting completely different coefficients.

Second Most Common Error:

Poor VISUALIZE organization: Not properly grouping like terms before combining

Students might combine terms haphazardly without systematic organization:

• Combining x² with x terms
• Missing terms during the combination process
• Getting confused about which terms can be combined

This leads to coefficient errors and may result in selecting Choice C (\(\mathrm{x}^2 + \mathrm{x} - 13\)) where the middle term sign is wrong.

The Bottom Line:

This problem tests careful execution of fundamental algebra skills. Success requires methodical distribution of negative coefficients and systematic organization when combining multiple like terms.