Which expression is equivalent to 9x^2 + 5x?

1. INFER the solution strategy

• Looking at \(9\mathrm{x}^2 + 5\mathrm{x}\), I notice both terms contain the variable x
• The best approach is to factor out the common factor
• This will simplify the expression into a product form

2. SIMPLIFY by factoring out the common factor

• Identify what's common: both \(9\mathrm{x}^2\) and \(5\mathrm{x}\) contain x
• Rewrite each term showing the common factor:
  • \(9\mathrm{x}^2 = \mathrm{x} \times 9\mathrm{x}\)
  • \(5\mathrm{x} = \mathrm{x} \times 5\)
• Factor out x: \(9\mathrm{x}^2 + 5\mathrm{x} = \mathrm{x}(9\mathrm{x} + 5)\)

3. INFER the correct answer choice

• My factored form \(\mathrm{x}(9\mathrm{x} + 5)\) matches Choice A exactly

Answer: A. x(9x + 5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students attempt to factor but extract different amounts from each term incorrectly. For example, they might try to factor out \(9\mathrm{x}\) from the first term and just \(\mathrm{x}\) from the second term, leading to something like \(9\mathrm{x}(\mathrm{x}) + \mathrm{x}(5)\), which they can't simplify further. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that factoring is needed and instead try other algebraic manipulations or simply guess without a systematic approach. They might also expand the answer choices incorrectly when trying to verify, leading them to select Choice B (\(5\mathrm{x}(9\mathrm{x} + 1)\)) or Choice C (\(9\mathrm{x}(\mathrm{x} + 5)\)) if their expansion contains errors.

The Bottom Line:

Success on this problem requires recognizing the factoring pattern and cleanly executing the algebraic steps. The key insight is seeing that \(\mathrm{x}\) appears in both terms and can be factored out completely.