To simplify the given rational expression
[tex]\[
\frac{6 x^2-24 x-192}{x^2+9 x+20}
\][/tex]
we shall follow these steps:
1. Factor the numerator and the denominator:
First, let's start with the numerator [tex]\(6 x^2 - 24 x - 192\)[/tex].
- Notice that we can factor out a common factor of 6:
[tex]\[
6(x^2 - 4x - 32)
\][/tex]
- Next, we need to factor [tex]\(x^2 - 4x - 32\)[/tex]. We look for two numbers that multiply to [tex]\(-32\)[/tex] and add to [tex]\(-4\)[/tex].
Those numbers are [tex]\(-8\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[
x^2 - 4x - 32 = (x - 8)(x + 4)
\][/tex]
- Therefore, the factored form of the numerator is:
[tex]\[
6(x - 8)(x + 4)
\][/tex]
Now, let's factor the denominator [tex]\(x^2 + 9 x + 20\)[/tex].
- We need to find two numbers that multiply to [tex]\(20\)[/tex] and add to [tex]\(9\)[/tex].
Those numbers are [tex]\(4\)[/tex] and [tex]\(5\)[/tex]:
[tex]\[
x^2 + 9x + 20 = (x + 4)(x + 5)
\][/tex]
2. Write the rational expression with the factored forms:
[tex]\[
\frac{6(x - 8)(x + 4)}{(x + 4)(x + 5)}
\][/tex]
3. Cancel common factors:
Here we can see that [tex]\((x + 4)\)[/tex] is a common factor in both the numerator and denominator:
[tex]\[
\frac{6(x - 8)(x + 4)}{(x + 4)(x + 5)} = \frac{6(x - 8)}{x + 5}
\][/tex]
Thus, the simplified form of the rational expression is:
[tex]\[
\frac{6(x - 8)}{x + 5}
\][/tex]
Therefore, our final simplified expression is:
[tex]\[
6 \cdot \frac{(x - 8)}{(x + 5)}
\][/tex]
[tex]\[
\boxed{\frac{6(x - 8)}{x + 5}}
\][/tex]
