Which expression is equivalent to x^2 + 3x - 40?

Step-by-Step Solution

1. TRANSLATE the problem information

• Given: \(\mathrm{x^2 + 3x - 40}\)
• Need: Equivalent factored expression from the choices
• This is a quadratic in standard form: \(\mathrm{x^2 + bx + c}\) where \(\mathrm{b = 3}\), \(\mathrm{c = -40}\)

2. INFER the factoring strategy

• For \(\mathrm{x^2 + bx + c}\), we need two numbers that:
  • Multiply to give \(\mathrm{c = -40}\)
  • Add to give \(\mathrm{b = 3}\)
• Since c is negative \(\mathrm{(-40)}\), our two numbers must have opposite signs
• Since b is positive \(\mathrm{(3)}\), the larger absolute value number must be positive

3. SIMPLIFY by finding the correct number pair

• Test factor pairs of -40:
  • 1 and -40: \(\mathrm{1 + (-40) = -39}\) ❌
  • -1 and 40: \(\mathrm{-1 + 40 = 39}\) ❌
  • 4 and -10: \(\mathrm{4 + (-10) = -6}\) ❌
  • -4 and 10: \(\mathrm{-4 + 10 = 6}\) ❌
  • 5 and -8: \(\mathrm{5 + (-8) = -3}\) ❌
  • -5 and 8: \(\mathrm{-5 + 8 = 3}\) ✓

4. APPLY CONSTRAINTS to write the factored form

• With numbers -5 and 8, the factored form is: \(\mathrm{(x - 5)(x + 8)}\)
• Verify: \(\mathrm{(x - 5)(x + 8) = x^2 + 8x - 5x - 40 = x^2 + 3x - 40}\) ✓

Answer: B. (x - 5)(x + 8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students get confused about the relationship between the signs in the factored form and the signs of the numbers they found. They might correctly find -5 and 8, but write \(\mathrm{(x + 5)(x - 8)}\) instead of \(\mathrm{(x - 5)(x + 8)}\).

When they expand \(\mathrm{(x + 5)(x - 8)}\), they get \(\mathrm{x^2 - 8x + 5x - 40 = x^2 - 3x - 40}\), which matches Choice C. The sign error in the middle term \(\mathrm{(-3x}\) instead of \(\mathrm{+3x)}\) comes from switching the signs in the binomial factors.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the wrong factor pair by mixing up what should multiply versus what should add. They might find 5 and 8 (which multiply to 40, not -40) and create \(\mathrm{(x + 5)(x + 8)}\), leading to \(\mathrm{x^2 + 13x + 40}\) instead of the original expression.

This leads to confusion since none of the answer choices match, causing them to guess randomly.

The Bottom Line:

Success requires systematically testing factor pairs while carefully tracking signs, then correctly translating those numbers into binomial factors with appropriate signs.