Simplify:
`\frac{x^{2}-2x-24}{x^{2}+7x+12}\cdot\frac{x^{2}-1}{x-6}`
To simplify this expression, we will first factor the quadratic expressions in both the numerator and the denominator:
For the numerator:
x^2 - 2x - 24 can be factored as (x - 6)(x + 4)
For the denominator:
x^2 + 7x + 12 can be factored as (x + 3)(x + 4)
Now, rewrite the original expression using these factorizations:
((x - 6)(x + 4))/(x + 3)(x + 4) * (x^2 - 1)/(x - 6)
Since (x + 4) and (x - 6) cancel out, we have:
= (x - 6)(x^2 - 1)/(x + 3)
Now, we can further simplify the numerator by expanding the expression x^2 - 1:
= (x - 6)(x^2 - 1) = (x - 6)(x + 1)(x - 1) = (x^2 - 6x + x - 6) = (x^2 - 5x - 6)
Therefore, the simplified expression is:
= (x^2 - 5x - 6)/(x + 3)