The expression x^2 + px + 120 can be factored as \(\mathrm{(x + m)(x + n)}\), where m, n, and...

1. TRANSLATE the factored form information

• Given information:
  • \(\mathrm{x^2 + px + 120 = (x + m)(x + n)}\)
  • \(\mathrm{m, n, p}\) are positive integers with \(\mathrm{m \lt n}\)
  • \(\mathrm{n - m = 7}\)

• What this tells us: We need to find relationships between \(\mathrm{m, n,}\) and \(\mathrm{p}\).

2. INFER the coefficient relationships

• Expand \(\mathrm{(x + m)(x + n)}\):

\(\mathrm{(x + m)(x + n) = x^2 + (m + n)x + mn}\)

• Compare with \(\mathrm{x^2 + px + 120}\):
  • Coefficient of \(\mathrm{x}\): \(\mathrm{p = m + n}\)
  • Constant term: \(\mathrm{mn = 120}\)

3. INFER the substitution strategy

• Since \(\mathrm{n - m = 7}\), we can write: \(\mathrm{n = m + 7}\)
• This lets us eliminate \(\mathrm{n}\) and solve for \(\mathrm{m}\) first
• Substitute into \(\mathrm{mn = 120}\): \(\mathrm{m(m + 7) = 120}\)

4. SIMPLIFY the quadratic equation

• Expand: \(\mathrm{m^2 + 7m = 120}\)
• Rearrange: \(\mathrm{m^2 + 7m - 120 = 0}\)
• Factor the quadratic: We need two numbers that multiply to \(\mathrm{-120}\) and add to \(\mathrm{7}\)
• Those numbers are \(\mathrm{15}\) and \(\mathrm{-8}\): \(\mathrm{(m + 15)(m - 8) = 0}\)

5. APPLY CONSTRAINTS to select the valid solution

• The quadratic gives: \(\mathrm{m = -15}\) or \(\mathrm{m = 8}\)
• Since \(\mathrm{m}\) must be positive: \(\mathrm{m = 8}\)
• Therefore: \(\mathrm{n = m + 7 = 15}\)
• Finally: \(\mathrm{p = m + n = 8 + 15 = 23}\)

Answer: 23

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often struggle to connect the constraint \(\mathrm{n - m = 7}\) with the equations \(\mathrm{p = m + n}\) and \(\mathrm{mn = 120}\). They might try to solve the system without substitution, leading to complex algebra with two variables. This approach becomes unwieldy and often causes them to make computational errors or abandon the systematic approach, leading to confusion and guessing.

Second Most Common Error:

Missing APPLY CONSTRAINTS reasoning: Some students correctly solve \(\mathrm{m^2 + 7m - 120 = 0}\) to get \(\mathrm{m = -15}\) or \(\mathrm{m = 8}\), but fail to recognize that \(\mathrm{m}\) must be positive. They might use \(\mathrm{m = -15}\), giving \(\mathrm{n = -8}\), which violates both the positivity constraint and \(\mathrm{m \lt n}\). This leads them to calculate \(\mathrm{p = (-15) + (-8) = -23}\), which isn't among the choices, causing them to get stuck and guess.

The Bottom Line:

This problem tests whether students can systematically use substitution to reduce a multi-variable constraint problem to a single-variable quadratic, then apply given constraints to select the correct solution.