The length of a rectangle is \(\mathrm{(x + 8)}\) units and the width is \(\mathrm{(x - 5)}\) units. Which of...

1. TRANSLATE the problem information

• Given information:
  • Length of rectangle: \(\mathrm{(x + 8)}\) units
  • Width of rectangle: \(\mathrm{(x - 5)}\) units
  • Need to find: Area of the rectangle

• Since area of rectangle = length × width, we need to calculate:
  \(\mathrm{Area = (x + 8)(x - 5)}\)

2. SIMPLIFY by expanding the product

• We need to multiply these two binomials: \(\mathrm{(x + 8)(x - 5)}\)
• Use FOIL method:
  • First terms: \(\mathrm{x × x = x²}\)
  • Outer terms: \(\mathrm{x × (-5) = -5x}\)
  • Inner terms: \(\mathrm{8 × x = 8x}\)
  • Last terms: \(\mathrm{8 × (-5) = -40}\)

3. SIMPLIFY by combining like terms

• From FOIL: \(\mathrm{x² + (-5x) + 8x + (-40)}\)
• Combine the x terms: \(\mathrm{-5x + 8x = 3x}\)
• Final expression: \(\mathrm{x² + 3x - 40}\)

Answer: B \(\mathrm{(x² + 3x - 40)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors during expansion, particularly with negative terms.

Many students correctly set up \(\mathrm{(x + 8)(x - 5)}\) but then make mistakes like:

• Forgetting the negative sign: calculating \(\mathrm{8 × 5 = 40}\) instead of \(\mathrm{8 × (-5) = -40}\)
• Sign errors in combining: getting \(\mathrm{-5x + 8x = -3x}\) instead of \(\mathrm{+3x}\)
• FOIL order confusion: mixing up outer and inner terms

This may lead them to select Choice A \(\mathrm{(x² + 13x - 40)}\) or Choice D \(\mathrm{(x² - 13x - 40)}\).

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students recognize they need length × width but struggle to organize the multiplication properly.

Some students might try to distribute incorrectly or add the expressions instead of multiplying them, leading to expressions that don't match any answer choice. This causes them to get stuck and guess.

The Bottom Line:

The algebraic expansion requires careful attention to signs and systematic application of FOIL. Students who rush through the multiplication or aren't methodical about tracking negative signs will struggle to arrive at the correct expression.