Questions: Add the following rational expressions and simplify your answer. -2x^2-x-1/(x-3) + 3x^2-5/(x-3) Give the simplified result for your answer. For example, if you found that the result was 1/(x+1), you would enter 1/(x+1).

Add the following rational expressions and simplify your answer.
-2x^2-x-1/(x-3) + 3x^2-5/(x-3)

Give the simplified result for your answer. For example, if you found that the result was 1/(x+1), you would enter 1/(x+1).

Transcript text: Question Add the following rational expressions and simplify your answer. \[ \frac{-2 x^{2}-x-1}{x-3}+\frac{3 x^{2}-5}{x-3} \] Give the simplified result for your answer. For example, if you found that the result was $\frac{1}{x+1}$, you would enter $\frac{1}{x+1}$. Provide your answer below:

Solution

Solution Steps

To add the given rational expressions, we first note that they have a common denominator. We can combine the numerators directly and then simplify the resulting expression.

Step 1: Combine the Numerators

Given the rational expressions: \[ \frac{-2x^2 - x - 1}{x - 3} + \frac{3x^2 - 5}{x - 3} \] Since the denominators are the same, we can combine the numerators: \[ \frac{-2x^2 - x - 1 + 3x^2 - 5}{x - 3} \]

Step 2: Simplify the Numerator

Combine like terms in the numerator: \[ -2x^2 + 3x^2 - x - 1 - 5 = x^2 - x - 6 \] So the expression becomes: \[ \frac{x^2 - x - 6}{x - 3} \]

Step 3: Factor the Numerator

Factor the quadratic expression $x^2 - x - 6$: \[ x^2 - x - 6 = (x - 3)(x + 2) \] Thus, the expression is: \[ \frac{(x - 3)(x + 2)}{x - 3} \]

Step 4: Simplify the Expression

Cancel the common factor $(x - 3)$ in the numerator and the denominator: \[ \frac{(x - 3)(x + 2)}{x - 3} = x + 2 \]