Sure, let's simplify the fraction [tex]\(\frac{x^2 + 5x + 6}{x^2 - 9}\)[/tex] step by step.
1. Factor the numerator and the denominator:
The numerator is [tex]\(x^2 + 5x + 6\)[/tex].
- This quadratic expression can be factored into [tex]\((x + 2)(x + 3)\)[/tex].
The denominator is [tex]\(x^2 - 9\)[/tex].
- This is a difference of squares and can be factored into [tex]\((x - 3)(x + 3)\)[/tex].
2. Rewrite the fraction using these factorizations:
So, [tex]\[\frac{x^2 + 5x + 6}{x^2 - 9} = \frac{(x + 2)(x + 3)}{(x - 3)(x + 3)}\][/tex]
3. Simplify the fraction by canceling common factors:
We see that [tex]\((x + 3)\)[/tex] is a common factor in both the numerator and the denominator.
- Therefore, we can cancel out [tex]\((x + 3)\)[/tex] from both.
So, the simplified fraction is: [tex]\[\frac{(x + 2)}{(x - 3)}\][/tex]
Therefore, the simplified form of the fraction [tex]\(\frac{x^2 + 5x + 6}{x^2 - 9}\)[/tex] is [tex]\(\frac{x + 2}{x - 3}\)[/tex].
Note that the domain restrictions for the variable [tex]\(x\)[/tex] must be considered. Specifically, [tex]\(x \neq -3\)[/tex] and [tex]\(x \neq 3\)[/tex] to avoid division by zero in the original fraction.
