Which expression is equivalent to \(-3(2\mathrm{x}^2 - 5)\)?

1. INFER the approach needed

• We need to eliminate the parentheses by distributing the -3 to each term inside
• This requires using the distributive property: \(\mathrm{a(b + c) = ab + ac}\)

2. SIMPLIFY by applying the distributive property

• Multiply -3 by the first term: \(\mathrm{(-3)(2x^2) = -6x^2}\)
• Multiply -3 by the second term: \(\mathrm{(-3)(-5) = +15}\)
• Combine the results: \(\mathrm{-6x^2 + 15}\)

Answer: B. \(\mathrm{-6x^2 + 15}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly handle the signs when multiplying \(\mathrm{(-3)(-5)}\)

Many students remember that "negative times negative equals positive" but get confused in the moment and write \(\mathrm{(-3)(-5) = -15}\) instead of \(\mathrm{+15}\). This leads them to get \(\mathrm{-6x^2 - 15}\).

This may lead them to select Choice A (\(\mathrm{-6x^2 - 15}\))

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly distribute to the first term but forget to distribute to the second term

Some students multiply -3 by \(\mathrm{2x^2}\) correctly to get \(\mathrm{-6x^2}\), but then mistakenly think they just subtract 5, writing \(\mathrm{-6x^2 - 5}\) instead of properly distributing -3 to the \(\mathrm{-5}\).

This may lead them to select Choice C (\(\mathrm{-6x^2 - 5}\))

The Bottom Line:

This problem tests careful execution of the distributive property, especially with negative signs. The key is systematically multiplying the outside factor by each term inside the parentheses and being extra careful with sign rules.