select the correct answer.\nwhat is this expression in simplest form?\n$\frac{x + 2}{4x^{2}+5x +…

select the correct answer.\nwhat is this expression in simplest form?\n$\frac{x + 2}{4x^{2}+5x + 1}cdot\frac{4x + 1}{x^{2}-4}$\na. $\frac{1}{(x + 1)(x - 2)}$\nb. $\frac{x}{(x - 2)}$\nc. $\frac{4x + 1}{(x + 1)(x - 2)}$\nd. $\frac{4x + 1}{x - 2}$

select the correct answer.\nwhat is this expression in simplest form?\n$\frac{x + 2}{4x^{2}+5x + 1}cdot\frac{4x + 1}{x^{2}-4}$\na. $\frac{1}{(x + 1)(x - 2)}$\nb. $\frac{x}{(x - 2)}$\nc. $\frac{4x + 1}{(x + 1)(x - 2)}$\nd. $\frac{4x + 1}{x - 2}$

Answer

Explanation:

Step1: Factor the denominators

Factor $4x^{2}+5x + 1=(4x + 1)(x+1)$ and $x^{2}-4=(x + 2)(x - 2)$ using the formula $a^{2}-b^{2}=(a + b)(a - b)$ for $x^{2}-4$ and factoring quadratic $4x^{2}+5x + 1$. The original expression becomes $\frac{x + 2}{(4x + 1)(x + 1)}\cdot\frac{4x+1}{(x + 2)(x - 2)}$.

Step2: Cancel out common factors

Cancel out the common factors $(x + 2)$ and $(4x + 1)$ in the numerator and denominator. We get $\frac{1}{(x + 1)(x - 2)}$.

Answer:

A. $\frac{1}{(x + 1)(x - 2)}$