Question

which expression is equal to \\(\frac{2x}{x - 2} - \frac{x + 3}{x + 5}\\)? \\(\bigcirc\\) \\(\frac{3x^2 + 11x + 6}{(x - 2)(x + 5)}\\) \\(\bigcirc\\) \\(\frac{x^2 + 8x + 10}{(x - 2)(x + 5)}\\) \\(\bigcirc\\) \\(\frac{x^2 + 9x + 6}{(x - 2)(x + 5)}\\) \\(\bigcirc\\) \\(\frac{x^2 + 11x - 6}{(x - 2)(x + 5)}\\)

Explanation:

Step1: Find common denominator

The common denominator is $(x-2)(x+5)$. Rewrite each fraction:
$$\frac{2x}{x-2} = \frac{2x(x+5)}{(x-2)(x+5)}, \quad \frac{x+3}{x+5} = \frac{(x+3)(x-2)}{(x-2)(x+5)}$$

Step2: Expand numerators

Calculate the expanded form of each numerator:
$$2x(x+5) = 2x^2 + 10x$$
$$(x+3)(x-2) = x^2 -2x +3x -6 = x^2 +x -6$$

Step3: Subtract the fractions

Subtract the second numerator from the first, over the common denominator:
$$\frac{(2x^2 + 10x) - (x^2 +x -6)}{(x-2)(x+5)}$$

Step4: Simplify the numerator

Combine like terms in the numerator:
$$2x^2 +10x -x^2 -x +6 = x^2 +9x +6$$

Answer:

$\boldsymbol{\frac{x^2+9x+6}{(x-2)(x+5)}}$ (the third option)