Question

select the correct answer.

what is this expression in simplest form?

\\\frac{x+2}{4x^2+5x+1} \cdot \frac{4x+1}{x^2-4}\\

• \\(\frac{1}{(x+1)(x-2)}\\)
• \\(\frac{x}{(x-2)}\\)
• \\(\frac{4x+1}{(x+1)(x-2)}\\)
• \\(\frac{4x+1}{x-2}\\)

Explanation:

Factor the denominators and numerators of the rational expressions

$$
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$$

Substitute the factored forms into the multiplication expression

$$ \frac{x + 2}{(4x + 1)(x + 1)} \cdot \frac{4x + 1}{(x - 2)(x + 2)} $$

Cancel common factors to simplify the expression

$$ \frac{\cancel{x + 2}}{\cancel{(4x + 1)}(x + 1)} \cdot \frac{\cancel{4x + 1}}{(x - 2)\cancel{(x + 2)}} = \frac{1}{(x + 1)(x - 2)} $$

Answer:

• (A) \(\frac{1}{(x + 1)(x - 2)}\) (Correct answer)
• (B) \(\frac{x}{(x - 2)}\)
• (C) \(\frac{4x + 1}{(x + 1)(x - 2)}\)
• (D) \(\frac{4x + 1}{x - 2}\)