Question

what is the difference?
\\(\frac{x}{x^2 + 3x + 2} - \frac{1}{(x + 2)(x + 1)}\\)
\\(\circ \frac{x - 1}{6x + 4}\\)
\\(\circ \frac{-1}{4x + 2}\\)
\\(\circ \frac{1}{x + 2}\\)
\\(\circ \frac{x - 1}{x^2 + 3x + 2}\\)

Explanation:

Step1: Factor the denominator

Factor \(x^2 + 3x + 2\). We know that \(x^2 + 3x + 2=(x + 1)(x + 2)\) (since \(1\times2 = 2\) and \(1+2=3\)). So the first fraction \(\frac{x}{x^2+3x + 2}\) becomes \(\frac{x}{(x + 1)(x + 2)}\), and the second fraction is \(\frac{1}{(x + 2)(x + 1)}\).

Step2: Subtract the fractions

Since the denominators are the same \((x + 1)(x + 2)\), we subtract the numerators: \(\frac{x}{(x + 1)(x + 2)}-\frac{1}{(x + 1)(x + 2)}=\frac{x-1}{(x + 1)(x + 2)}\). But \((x + 1)(x + 2)=x^2+3x + 2\), so the result is \(\frac{x - 1}{x^2+3x + 2}\).

Answer:

\(\frac{x-1}{x^2+3x+2}\) (the fourth option, D. \(\frac{x - 1}{x^2+3x + 2}\) assuming the options are labeled A, B, C, D in order)