Question

011 algebra 2 mp3 6 weeks 20-26
$\frac{x - 2}{x^2 + 5x + 6} - \frac{5x + 10}{x^2 + 5x + 6}$
options:
$\bigcirc -\frac{4}{x+2}$
$\bigcirc -\frac{4x - 12}{(x+3)(x+2)}$
$\bigcirc \frac{4}{x+2}$
$\bigcirc \frac{4x - 12}{(x+3)(x+2)}$
clear all

Explanation:

Step1: Factor denominators

First, factor the quadratic denominators:
$x^2 + 5x + 6 = (x+2)(x+3)$
$5x + 10 = 5(x+2)$
The expression becomes:
$\frac{x-2}{(x+2)(x+3)} - \frac{5(x+2)}{(x+2)(x+3)}$

Step2: Combine the fractions

Since the denominators are now the same, subtract the numerators:
$\frac{(x-2) - 5(x+2)}{(x+2)(x+3)}$

Step3: Simplify the numerator

Expand and combine like terms in the numerator:
$(x-2) -5x -10 = x -2 -5x -10 = -4x -12 = -4(x+3)$
The expression is now:
$\frac{-4(x+3)}{(x+2)(x+3)}$

Step4: Cancel common factors

Cancel the $(x+3)$ term from numerator and denominator (where $x
eq -3$):
$\frac{-4}{x+2}$

Answer:

$\boldsymbol{-\frac{4}{x+2}}$ (matches the first option: $\boldsymbol{-\frac{4}{x+2}}$)