Answer
The domain of this function is all real numbers except $-3$ and $-2$.
Work Step by Step
To find the domain of this function, we need to find which values are excluded for $r$. In a rational function, the denominator cannot equal $0$ because the function would be undefined. Therefore, we need to set the denominator equal to $0$ and solve for $r$:
$r^2 + 5r + 6 = 0$
We can factor this equation to solve for $r$.
We see that we have a quadratic equation, which is given by the formula:
$ax^2 + bx + c$, where $a$, $b$, and $c$ are all real numbers.
To factor this equation, we want to find which factors when multiplied will give us the product of the $a$ and $c$ terms, which is $6$, but when added together will give us the $b$ term, which is $5$. This means that both factors should be positive.
Let's look at possible factors:
$3$ and $2$
$6$ and $-1$
It looks like the first combination will work. Let's put the factors together:
$(r + 3)(r + 2) = 0$
According to the zero product property, if the product of two factors equals $0$, then either factor can be $0$; therefore, we can set each of these factors equal to $0$ and solve:
$r + 3 = 0$ or $r + 2 = 0$
Add or subtract to solve:
$r = -3$ or $r = -2$
These are the numbers $r$ cannot be. Therefore, the domain of this function is all real numbers except $-3$ and $-2$.