Add the following rational expressions: 2x/x2+4x−5+1/x+5 . Enter response with terms arranged in order from largest degree to smallest degree.

Answer 1

To add the rational expressions \(\frac{2x}{x^2 + 4x - 5}\) and \(\frac{1}{x + 5}\), we first need to find a common denominator.

1. Factor the denominator of the first term: \[ x^2 + 4x - 5 = (x + 5)(x - 1) \] So we rewrite the first term: \[ \frac{2x}{(x + 5)(x - 1)} \]

2. Identify the common denominator: The common denominator between \((x + 5)(x - 1)\) and \((x + 5)\) is \((x + 5)(x - 1)\).

3. Rewriting the second term with the common denominator: \[ \frac{1}{x + 5} = \frac{1 \cdot (x - 1)}{(x + 5)(x - 1)} = \frac{x - 1}{(x + 5)(x - 1)} \]

4. Combine the two fractions: Now we can add the two fractions: \[ \frac{2x}{(x + 5)(x - 1)} + \frac{x - 1}{(x + 5)(x - 1)} = \frac{2x + (x - 1)}{(x + 5)(x - 1)} \]

5. Simplify the numerator: Combine like terms in the numerator: \[ 2x + (x - 1) = 2x + x - 1 = 3x - 1 \]

6. Write the final expression: Therefore, the sum of the rational expressions is: \[ \frac{3x - 1}{(x + 5)(x - 1)} \]

So the final answer, with terms arranged from largest degree to smallest degree, is: \[ \frac{3x - 1}{(x + 5)(x - 1)} \]