To simplify the fraction [tex]\(\frac{x^2 + 5x - 6}{x^2 + 9x + 18}\)[/tex], we need to factor both the numerator and the denominator.
First, let's factor the numerator:
The numerator is [tex]\(x^2 + 5x - 6\)[/tex]. To factor this quadratic expression, we need to find two numbers that multiply to [tex]\(-6\)[/tex] (the constant term) and add up to [tex]\(5\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
The numbers that satisfy these criteria are [tex]\(6\)[/tex] and [tex]\(-1\)[/tex]. Therefore, the numerator can be factored as:
[tex]\[
x^2 + 5x - 6 = (x - 1)(x + 6)
\][/tex]
Next, let's factor the denominator:
The denominator is [tex]\(x^2 + 9x + 18\)[/tex]. We need to find two numbers that multiply to [tex]\(18\)[/tex] (the constant term) and add up to [tex]\(9\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
The numbers that satisfy these criteria are [tex]\(3\)[/tex] and [tex]\(6\)[/tex]. Therefore, the denominator can be factored as:
[tex]\[
x^2 + 9x + 18 = (x + 3)(x + 6)
\][/tex]
Now, we can write the fraction with the factored forms of the numerator and denominator:
[tex]\[
\frac{x^2 + 5x - 6}{x^2 + 9x + 18} = \frac{(x - 1)(x + 6)}{(x + 3)(x + 6)}
\][/tex]
We notice that [tex]\((x + 6)\)[/tex] is a common factor in both the numerator and the denominator. We can cancel this common factor:
[tex]\[
\frac{(x - 1)(x + 6)}{(x + 3)(x + 6)} = \frac{x - 1}{x + 3}
\][/tex]
Therefore, the simplest form of [tex]\(\frac{x^2 + 5x - 6}{x^2 + 9x + 18}\)[/tex] is:
[tex]\[
\frac{x - 1}{x + 3}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{x-1}{x+3}}
\][/tex]
