To solve the inequality $\frac{(x + 2)(x - 5)}{x + 4} \leq 0$ and write the solution in interval notation, we'll follow these steps:

Step 1: Identify the critical points

The critical points are the values of $x$ where the expression is equal to zero or undefined. These occur when the numerator or the denominator equals zero.

1. Numerator: $(x + 2)(x - 5) = 0$

   • $x + 2 = 0$ ⇒ $x = -2$
   • $x - 5 = 0$ ⇒ $x = 5$

2. Denominator: $x + 4 = 0$

   • $x = -4$

Step 2: Determine the sign of the expression on each interval

The critical points divide the number line into intervals. We test each interval to determine the sign of the expression.

The intervals to consider are:

• $(-\infty, -4)$
• $(-4, -2)$
• $(-2, 5)$
• $(5, \infty)$

Let's test a point from each interval:

1. Interval $(- \infty, -4)$, test with $x = -5$:

   • $\frac{(-5 + 2)(-5 - 5)}{-5 + 4} = \frac{(-3)(-10)}{-1} = \frac{30}{-1} = -30$
   • The expression is negative.

2. Interval $(-4, -2)$, test with $x = -3$:

   • $\frac{(-3 + 2)(-3 - 5)}{-3 + 4} = \frac{(-1)(-8)}{1} = \frac{8}{1} = 8$
   • The expression is positive.

3. Interval $(-2, 5)$, test with $x = 0$:

   • $\frac{(0 + 2)(0 - 5)}{0 + 4} = \frac{(2)(-5)}{4} = \frac{-10}{4} = -2.5$
   • The expression is negative.

4. Interval $(5, \infty)$, test with $x = 6$:

   • $\frac{(6 + 2)(6 - 5)}{6 + 4} = \frac{(8)(1)}{10} = \frac{8}{10} = 0.8$
   • The expression is positive.

Step 3: Consider the equality and vertical asymptote

• The expression equals zero at $x = -2$ and $x = 5$.
• The expression is undefined at $x = -4$ (vertical asymptote).

Step 4: Write the solution in interval notation

We are looking for where the expression is less than or equal to zero:

• On the intervals $(- \infty, -4)$ and $(-2, 5)$, the expression is negative.
• Include the points where the expression is equal to zero: $x = -2$ and $x = 5$.

Final solution: $(-\infty, -4) \cup [-2, 5]$

Would you like more details on any of the steps or have any questions?

Here are five related questions that expand on this topic:

1. How do you solve inequalities involving more complex rational expressions?
2. What happens when the numerator and denominator both contain quadratic expressions?
3. How do vertical asymptotes influence the solution set in inequalities?
4. What is the difference between strict inequalities and non-strict inequalities in terms of solution sets?
5. How do you verify your solution to an inequality using a graph?

Tip: Always check the points where the expression is undefined, as they could be important for determining the solution intervals.