The expression 4x^2 + bx - 45, where b is a constant, can be rewritten as \((\mathrm{hx} + \mathrm{k})(\mathrm{x} +...

1. TRANSLATE the problem information

• Given information:
  • \(4\mathrm{x}^2 + \mathrm{bx} - 45\) can be written as \((\mathrm{hx} + \mathrm{k})(\mathrm{x} + \mathrm{j})\)
  • h, k, and j are integer constants
  • Need to find which expression must be an integer

2. SIMPLIFY by expanding the factored form

• Expand \((\mathrm{hx} + \mathrm{k})(\mathrm{x} + \mathrm{j})\):

\((\mathrm{hx} + \mathrm{k})(\mathrm{x} + \mathrm{j}) = \mathrm{hx}^2 + \mathrm{jhx} + \mathrm{kx} + \mathrm{kj}\)

\(= \mathrm{hx}^2 + (\mathrm{jh} + \mathrm{k})\mathrm{x} + \mathrm{kj}\)

3. INFER coefficient relationships

• Since \(4\mathrm{x}^2 + \mathrm{bx} - 45 = \mathrm{hx}^2 + (\mathrm{jh} + \mathrm{k})\mathrm{x} + \mathrm{kj}\), the coefficients must match:
  • x² coefficient: \(\mathrm{h} = 4\)
  • x coefficient: \(\mathrm{b} = \mathrm{jh} + \mathrm{k}\)
  • constant term: \(\mathrm{kj} = -45\)

4. INFER which expression must be an integer

• From \(\mathrm{kj} = -45\) and the constraint that j is an integer:

\(\mathrm{j} = \frac{-45}{\mathrm{k}}\)

• Since j must be an integer, \(\frac{-45}{\mathrm{k}}\) must be an integer
• Therefore, \(\frac{45}{\mathrm{k}}\) must also be an integer

5. Check the answer choices

• A. \(\frac{\mathrm{b}}{\mathrm{h}}\): We don't have enough constraints to guarantee this is an integer
• B. \(\frac{\mathrm{b}}{\mathrm{k}}\): Similarly, no guarantee this is an integer
• C. \(\frac{45}{\mathrm{h}} = \frac{45}{4} = 11.25\): Clearly not an integer
• D. \(\frac{45}{\mathrm{k}}\): Must be an integer (proven above)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students expand the factored form correctly but fail to systematically compare coefficients to establish the key relationships \(\mathrm{h} = 4\), \(\mathrm{b} = \mathrm{jh} + \mathrm{k}\), and \(\mathrm{kj} = -45\).

Instead, they might try to work backwards from the answer choices or attempt to factor \(4\mathrm{x}^2 + \mathrm{bx} - 45\) directly without using the given constraint that it equals \((\mathrm{hx} + \mathrm{k})(\mathrm{x} + \mathrm{j})\). This leads to confusion about which relationships are actually required, causing them to guess among the choices.

Second Most Common Error:

Missing conceptual understanding of integer constraints: Students correctly identify that \(\mathrm{kj} = -45\) but don't recognize the implication that since j must be an integer, \(\frac{45}{\mathrm{k}}\) must also be an integer.

They might incorrectly assume that since k and j are integers, all the expressions in the answer choices should be integers, leading them to select Choice A or Choice B without proper justification.

The Bottom Line:

This problem tests whether students can systematically use coefficient comparison and apply integer constraints logically. The key insight is recognizing that the integer requirement on j creates a necessary divisibility condition for \(\frac{45}{\mathrm{k}}\).