To determine the domain of the given function:

$f(x) = \frac{\sqrt{x^2 + 3x + 2}}{x - 5}$

Steps to find the domain:

1. Numerator Restriction:
   The square root, $\sqrt{x^2 + 3x + 2}$, requires that the expression inside the square root is non-negative: $x^2 + 3x + 2 \geq 0$

   Factorize the quadratic: $x^2 + 3x + 2 = (x + 1)(x + 2)$

   The inequality becomes: $(x + 1)(x + 2) \geq 0$

   Solve this using a sign chart:

   • The critical points are $x = -2$ and $x = -1$.
   • Test intervals $(- \infty, -2)$, $(-2, -1)$, and $(-1, \infty)$:
     • $(x + 1)(x + 2) > 0$ for $x \in (-\infty, -2] \cup [-1, \infty)$.

2. Denominator Restriction:
   The denominator $x - 5 \neq 0$, so: $x \neq 5$

3. Combine Restrictions:
   The domain is: $(-\infty, -2] \cup [-1, 5) \cup (5, \infty)$

Match with the options:

The domain corresponds to option (a): $\mathbb{R} - ([-2, -1] \cup \{5\})$.

Would you like a deeper explanation of any step?

5 Related Questions:

1. How do you solve quadratic inequalities in general?
2. What is the significance of excluding $x = 5$ in this problem?
3. Can this function ever be undefined for other reasons?
4. How does the domain of a square root function differ from that of a rational function?
5. What are the visual implications of this domain on a graph?

Tip:

For inequalities involving products or divisions, always use a sign chart to carefully determine the solution set.