Which expression is equivalent to x^2 + 7x + 12?\(\mathrm{(x + 2)(x + 6)}\)\(\mathrm{(x + 3)(x + 4)}\)\(\mathrm{x}(\mathrm{x + 7})...

1. INFER the problem strategy

• The problem asks for an equivalent expression to \(\mathrm{x^2 + 7x + 12}\)
• Looking at the answer choices, most are factored forms, so I need to factor the given quadratic
• For \(\mathrm{x^2 + bx + c}\), I need two numbers that multiply to c and add to b

2. INFER the specific numbers needed

• I need two numbers that multiply to 12 (the constant term)
• These same numbers must add to 7 (the linear coefficient)
• Let me systematically check factor pairs of 12

3. SIMPLIFY by finding the factor pairs

• Factor pairs of 12:
  • 1 and 12: \(\mathrm{1 \times 12 = 12}\), but \(\mathrm{1 + 12 = 13 \neq 7}\)
  • 2 and 6: \(\mathrm{2 \times 6 = 12}\), but \(\mathrm{2 + 6 = 8 \neq 7}\)
  • 3 and 4: \(\mathrm{3 \times 4 = 12}\), and \(\mathrm{3 + 4 = 7}\) ✓

4. SIMPLIFY the factorization

• Using 3 and 4: \(\mathrm{x^2 + 7x + 12 = (x + 3)(x + 4)}\)

5. SIMPLIFY by verifying the answer

• Expand \(\mathrm{(x + 3)(x + 4)}\):
  \(\mathrm{(x + 3)(x + 4) = x^2 + 4x + 3x + 12}\)
  \(\mathrm{= x^2 + 7x + 12}\) ✓

Answer: B. (x + 3)(x + 4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may recognize they need to factor but incorrectly identify which two numbers to use. They might choose numbers that add to 12 and multiply to 7 (reversing the relationship), or randomly try factor pairs without systematic checking.

For example, they might try 1 and 12 because these "look related" to the coefficients, leading to \(\mathrm{(x + 1)(x + 12)}\), which expands to \(\mathrm{x^2 + 13x + 12}\).

This may lead them to select Choice D ((x + 1)(x + 12)).

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the factoring strategy but make arithmetic errors when checking factor pairs or when expanding for verification. They might incorrectly calculate that \(\mathrm{2 + 6 = 7}\) instead of 8.

This may lead them to select Choice A ((x + 2)(x + 6)).

The Bottom Line:

Success requires both strategic thinking (knowing the factor-pair relationship for quadratics) and careful arithmetic execution. The systematic approach of listing all factor pairs and checking both multiplication and addition is crucial for avoiding errors.