The given rational function is:

$g(x) = \frac{-5x - 20}{x + 4}$

Step 1: Simplify the Rational Function

We can factor the numerator:

$g(x) = \frac{-5(x + 4)}{x + 4}$

Since $x + 4$ appears in both the numerator and the denominator, it cancels out, but it introduces a restriction. The simplified function becomes:

$g(x) = -5$

However, there is a hole in the graph at $x = -4$, because $x + 4 = 0$ at this point and division by zero is undefined.

Step 2: Asymptotes

• Vertical Asymptote: Normally, we check where the denominator equals zero for vertical asymptotes. But here, the factor $x + 4$ cancels, and there is no vertical asymptote—just a hole at $x = -4$.
• Horizontal Asymptote: Since the simplified function is $g(x) = -5$, which is constant, the graph is just a horizontal line at $y = -5$.

Step 3: Plotting the Graph

• The graph is a horizontal line at $y = -5$, except there is a hole at $x = -4$.
• To graph this, plot a line through $y = -5$ and place a hollow dot at $(-4, -5)$.

Would you like a detailed step-by-step guide on how to graph this? Additionally, here are some related questions for you to explore:

1. What is a vertical asymptote, and how does it differ from a hole?
2. How do you find horizontal asymptotes for rational functions in general?
3. Can you cancel terms in rational functions and still have the same function?
4. What is the behavior of a function near a hole?
5. What is the significance of horizontal lines in function graphs?

Tip: When simplifying rational functions, always check for restrictions (like holes) that arise from canceled terms.