Which of the following is equivalent to the sum of \(\mathrm{x(2x^2 + 3x)}\) and 5x^3 + 4x^2?

1. INFER the approach needed

• This problem requires adding two polynomial expressions
• But first expression has a factor to distribute: \(\mathrm{x(2x^2 + 3x)}\)
• Strategy: Distribute first, then add the results

2. SIMPLIFY by distributing the x

• Distribute: \(\mathrm{x(2x^2 + 3x) = x \cdot 2x^2 + x \cdot 3x}\)
• Apply exponent rule: \(\mathrm{x \cdot x^2 = x^3}\)
• Result: \(\mathrm{x(2x^2 + 3x) = 2x^3 + 3x^2}\)

3. SIMPLIFY by adding the expressions

• Now add: \(\mathrm{(2x^3 + 3x^2) + (5x^3 + 4x^2)}\)
• Remove parentheses: \(\mathrm{2x^3 + 3x^2 + 5x^3 + 4x^2}\)

4. SIMPLIFY by combining like terms

• Combine x³ terms: \(\mathrm{2x^3 + 5x^3 = 7x^3}\)
• Combine x² terms: \(\mathrm{3x^2 + 4x^2 = 7x^2}\)
• Final result: \(\mathrm{7x^3 + 7x^2}\)

Answer: C (\(\mathrm{7x^3 + 7x^2}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution during distribution: Students incorrectly distribute \(\mathrm{x(2x^2 + 3x)}\), often getting \(\mathrm{2x^3 + 3x}\) instead of \(\mathrm{2x^3 + 3x^2}\).

They might think: "x times 3x equals 3x" instead of recognizing that \(\mathrm{x \cdot 3x = 3x^2}\).

When they continue with \(\mathrm{(2x^3 + 3x) + (5x^3 + 4x^2)}\) and combine like terms, they get \(\mathrm{7x^3 + 4x^2 + 3x}\).

This leads them to select Choice A (\(\mathrm{7x^3 + 4x^2 + 3x}\)).

The Bottom Line:

This problem tests whether students can correctly apply the distributive property with polynomial terms, particularly remembering that \(\mathrm{x \cdot x^n = x^{n+1}}\). The distribution step is where most errors occur, and these errors carry through to produce a plausible but incorrect final answer.