Question

q1 subtract the rational expressions and simplify.
\\(\frac{x}{x^2 + 5x + 6} - \frac{2}{x^2 - 9}\\)
\\(\bigcirc \frac{x^2 - 5x + 4}{x^3 + 2x^2 - 9x - 18}\\)
\\(\bigcirc \frac{x - 2}{5x - 3}\\)
\\(\bigcirc \frac{x^2 - 5x - 4}{x^3 + 2x^2 - 9x - 18}\\)
\\(\bigcirc \frac{x - 2}{5x + 15}\\)

Explanation:

Step1: Factor denominators

Factor \(x^2 + 5x + 6\) as \((x + 2)(x + 3)\) and \(x^2 - 9\) as \((x + 3)(x - 3)\). So the expression becomes \(\frac{x}{(x + 2)(x + 3)} - \frac{2}{(x + 3)(x - 3)}\).

Step2: Find common denominator

The common denominator is \((x + 2)(x + 3)(x - 3)\). Multiply numerator and denominator of each fraction: \(\frac{x(x - 3)}{(x + 2)(x + 3)(x - 3)} - \frac{2(x + 2)}{(x + 2)(x + 3)(x - 3)}\).

Step3: Subtract numerators

Subtract the numerators: \(x(x - 3) - 2(x + 2) = x^2 - 3x - 2x - 4 = x^2 - 5x - 4\)? Wait, no, wait. Wait, let's recalculate: \(x(x - 3)=x^2 - 3x\), \(2(x + 2)=2x + 4\), so subtracting: \(x^2 - 3x - (2x + 4)=x^2 - 3x - 2x - 4 = x^2 - 5x - 4\)? Wait, no, the original subtraction is \(\frac{x(x - 3)-2(x + 2)}{(x + 2)(x + 3)(x - 3)}\). Wait, but let's check the options. Wait, maybe I made a mistake. Wait, let's re - factor the denominator of the first option: \(x^3 + 2x^2 - 9x - 18\). Let's factor that: \(x^2(x + 2)-9(x + 2)=(x + 2)(x^2 - 9)=(x + 2)(x + 3)(x - 3)\), which is the common denominator. Now, let's recalculate the numerator: \(x(x - 3)-2(x + 2)=x^2 - 3x - 2x - 4=x^2 - 5x - 4\)? Wait, no, wait the first option's numerator is \(x^2 - 5x + 4\), the third is \(x^2 - 5x - 4\). Wait, maybe I messed up the sign. Wait, the subtraction is \(\frac{x(x - 3)}{denominator}-\frac{2(x + 2)}{denominator}=\frac{x(x - 3)-2(x + 2)}{denominator}\). So \(x(x - 3)=x^2 - 3x\), \(2(x + 2)=2x + 4\), so \(x^2 - 3x-(2x + 4)=x^2 - 3x - 2x - 4=x^2 - 5x - 4\). Wait, but the third option has numerator \(x^2 - 5x - 4\) and denominator \((x + 2)(x + 3)(x - 3)=x^3 + 2x^2 - 9x - 18\), which matches the third option. Wait, but let's check again. Wait, maybe I made a mistake in the sign when subtracting. Wait, the original problem is \(\frac{x}{x^2 + 5x + 6}-\frac{2}{x^2 - 9}\). So it's \(\frac{x}{(x + 2)(x + 3)}-\frac{2}{(x + 3)(x - 3)}\). The common denominator is \((x + 2)(x + 3)(x - 3)\). So first fraction: multiply numerator and denominator by \((x - 3)\): \(\frac{x(x - 3)}{(x + 2)(x + 3)(x - 3)}\). Second fraction: multiply numerator and denominator by \((x + 2)\): \(\frac{2(x + 2)}{(x + 2)(x + 3)(x - 3)}\). Now subtract the numerators: \(x(x - 3)-2(x + 2)=x^2 - 3x - 2x - 4=x^2 - 5x - 4\). And the denominator is \((x + 2)(x + 3)(x - 3)=x^3 + 2x^2 - 9x - 18\). So the expression is \(\frac{x^2 - 5x - 4}{x^3 + 2x^2 - 9x - 18}\), which is the third option.

Answer:

\(\boldsymbol{\frac{x^2 - 5x - 4}{x^3 + 2x^2 - 9x - 18}}\) (corresponding to the third option: \(\frac{x^2 - 5x - 4}{x^3 + 2x^2 - 9x - 18}\))