The problem asks to simplify the given rational expression: $\frac{2x^2 - 9x - 35}{x^3 + 2x^2 - 5x - 6}$

1. Problem Description

The problem asks to simplify the given rational expression:

$\frac{2x^2 - 9x - 35}{x^3 + 2x^2 - 5x - 6}$

2. Solution Steps

First, we factor the numerator. We look for two numbers that multiply to $2 \cdot (-35) = -70$ and add up to $-9$. These numbers are $-14$ and $5$. So we can rewrite the numerator as:

$2x^2 - 9x - 35 = 2x^2 - 14x + 5x - 35 = 2x(x - 7) + 5(x - 7) = (2x + 5)(x - 7)$

Now, we factor the denominator. We can try to find integer roots using the Rational Root Theorem. The possible integer roots are divisors of $-6$, which are $\pm 1, \pm 2, \pm 3, \pm 6$.

Let $P(x) = x^3 + 2x^2 - 5x - 6$.

$P(1) = 1 + 2 - 5 - 6 = -8 \neq 0$

$P(-1) = -1 + 2 + 5 - 6 = 0$, so $x = -1$ is a root. Thus, $(x + 1)$ is a factor.

We perform polynomial division to find the other factor:

$(x^3 + 2x^2 - 5x - 6) / (x + 1) = x^2 + x - 6$

So, $x^3 + 2x^2 - 5x - 6 = (x + 1)(x^2 + x - 6)$.

Now, we factor the quadratic $x^2 + x - 6$. We look for two numbers that multiply to $-6$ and add up to $1$. These numbers are $3$ and $-2$. So,

$x^2 + x - 6 = (x + 3)(x - 2)$

Therefore, $x^3 + 2x^2 - 5x - 6 = (x + 1)(x + 3)(x - 2)$

Now, we can write the rational expression as:

$\frac{(2x + 5)(x - 7)}{(x + 1)(x + 3)(x - 2)}$

There are no common factors between the numerator and denominator.

3. Final Answer

$\frac{(2x+5)(x-7)}{(x+1)(x+3)(x-2)}$