The problem asks to find the domain of the expression $\frac{x-1}{x^2 + 5x + 6}$. The domain of a rational function is all real numbers except for the values that make the denominator equal to zero.

1. Problem Description

The problem asks to find the domain of the expression $\frac{x-1}{x^2 + 5x + 6}$. The domain of a rational function is all real numbers except for the values that make the denominator equal to zero.

2. Solution Steps

To find the domain, we need to find the values of $x$ for which the denominator is not equal to zero.

We set the denominator equal to zero and solve for $x$:

$x^2 + 5x + 6 = 0$

We can factor the quadratic expression:

$(x+2)(x+3) = 0$

This gives us two possible solutions for $x$:

$x+2 = 0$ or $x+3 = 0$

$x = -2$ or $x = -3$

These values make the denominator zero, so they must be excluded from the domain. The domain is all real numbers except $x = -2$ and $x = -3$.

In interval notation, the domain is $(-\infty, -3) \cup (-3, -2) \cup (-2, \infty)$.

3. Final Answer

$(-\infty, -3) \cup (-3, -2) \cup (-2, \infty)$