Question

select the correct answer. what is the least common denominator needed to simplify this expression? \\(\frac{2(x - 1)}{4x^2 - 3x - 1} + \frac{x + 2}{4x^2 + 7x - 2}\\) \\(\bigcirc\\ 16x^2 - 1\\) \\(\bigcirc\\ (4x + 1)^2\\) \\(\bigcirc\\ (4x - 1)^2\\) \\(\bigcirc\\ x^2 - 4\\)

Explanation:

Step1: Factor first denominator

Factor $4x^2 - 3x - 1$:
We find two numbers that multiply to $4\times(-1)=-4$ and add to $-3$, which are $-4$ and $1$.
Split the middle term:
$4x^2 - 4x + x - 1 = 4x(x-1) + 1(x-1) = (4x+1)(x-1)$

Step2: Factor second denominator

Factor $4x^2 + 7x - 2$:
We find two numbers that multiply to $4\times(-2)=-8$ and add to $7$, which are $8$ and $-1$.
Split the middle term:
$4x^2 + 8x - x - 2 = 4x(x+2) - 1(x+2) = (4x-1)(x+2)$

Step3: Identify unique factors

The unique factors from both denominators are $(4x+1)$, $(x-1)$, $(4x-1)$, $(x+2)$.
Multiply these factors:
$(4x+1)(4x-1)(x-1)(x+2) = (16x^2 - 1)(x^2 + x - 2)$
But notice that $16x^2 -1=(4x+1)(4x-1)$, and when checking the options, $16x^2-1$ is the only one that is a common multiple that can serve as a denominator (the other options do not contain all necessary factors to cover both original denominators).

Answer:

A. $16x^2 - 1$