Question

find the sum: \\(\frac{x - 2}{x^2 + 1} + \frac{x + 3}{x^2 + 1}\\) options: \\(\frac{2x + 1}{2x^2 + 2}\\), \\(\frac{2x + 1}{x^2 + 1}\\), \\(\frac{1}{x^2 + 1}\\), \\(\frac{2}{x}\\)

Explanation:

Step1: Add the numerators (common denominator)

Since the denominators are the same ($x^2 + 1$), we add the numerators: $(x - 2) + (x + 3)$.
$= x - 2 + x + 3$

Step2: Combine like terms in numerator

Combine the $x$ terms and the constant terms: $x + x - 2 + 3 = 2x + 1$.

Step3: Write the sum over the common denominator

The denominator remains $x^2 + 1$, so the sum is $\frac{2x + 1}{x^2 + 1}$.

Answer:

$\frac{2x + 1}{x^2 + 1}$ (corresponding to the option with numerator $2x + 1$ and denominator $x^2 + 1$)