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 \((x+1)(x + 2)=x^2+2x+x + 2=x^2+3x + 2\). So the first fraction \(\frac{x}{x^2+3x + 2}\) can be written as \(\frac{x}{(x + 1)(x + 2)}\). The second fraction is \(\frac{1}{(x + 2)(x + 1)}\), so now both fractions have the same denominator \((x + 1)(x + 2)\).

Step2: Subtract the fractions

When subtracting fractions with the same denominator, we subtract the numerators and keep the denominator the same. So \(\frac{x}{(x + 1)(x + 2)}-\frac{1}{(x + 1)(x + 2)}=\frac{x-1}{(x + 1)(x + 2)}\). But \((x + 1)(x + 2)=x^2+3x + 2\), so the result is \(\frac{x - 1}{x^2+3x + 2}\).

Answer:

\(\frac{x-1}{x^2+3x + 2}\) (which corresponds to the option: \(\boldsymbol{\frac{x-1}{x^{2}+3x+2}}\))