Question

perform the indicated operations:
\\(\frac{2}{x - 2} + \frac{x}{x + 9} - \frac{x + 20}{x^2 + 7x - 18}\\)
\\(\circ\\) \\(\frac{x^2 - x + 2}{x^2 + 7x - 18}\\)
\\(\circ\\) \\(\frac{x + 1}{x + 9}\\)
\\(\circ\\) \\(\frac{22}{x^2 + 7x - 18}\\)
\\(\circ\\) \\(\frac{x^2 + x + 38}{x^2 + 7x - 18}\\)

Explanation:

Step1: Factor the quadratic denominator

$x^2+7x-18=(x-2)(x+9)$

Step2: Find common denominator

The common denominator is $(x-2)(x+9)$

Step3: Rewrite each fraction

$\frac{2(x+9)}{(x-2)(x+9)} + \frac{x(x-2)}{(x-2)(x+9)} - \frac{x+20}{(x-2)(x+9)}$

Step4: Combine numerators

$2(x+9)+x(x-2)-(x+20)$
$=2x+18+x^2-2x-x-20$

Step5: Simplify numerator

$x^2 - x - 2$

Step6: Factor simplified numerator

$x^2 - x - 2=(x-2)(x+1)$

Step7: Cancel common factors

$\frac{(x-2)(x+1)}{(x-2)(x+9)}=\frac{x+1}{x+9}$

Answer:

$\boldsymbol{\frac{x+1}{x+9}}$ (matches the second option: $\frac{x+1}{x+9}$)