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To solve the inequality \((k - 2)(k + 6) \leq k^2 + 24\), let’s go through the following steps:

### Step 1: Expand the Left Side
Expanding \((k - 2)(k + 6)\):
\[
(k - 2)(k + 6) = k^2 + 6k - 2k - 12 = k^2 + 4k - 12
\]

### Step 2: Set Up the Inequality
Now, substitute the expanded form into the inequality:
\[
k^2 + 4k - 12 \leq k^2 + 24
\]

### Step 3: Simplify the Inequality
Subtract \(k^2\) from both sides to simplify:
\[
4k - 12 \leq 24
\]

### Step 4: Solve for \(k\)
Add 12 to both sides:
\[
4k \leq 36
\]
Now, divide by 4:
\[
k \leq 9
\]

### Final Answer
The solution to the inequality is:
\[
k \leq 9
\]

Would you like more details on any part of this solution or have any other questions?

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Here are 5 related questions to deepen your understanding:

1. How would the solution change if the inequality were \((k - 2)(k + 6) \geq k^2 + 24\)?
2. What steps are necessary to solve a quadratic inequality involving terms with \(k^2\) on both sides?
3. Can you explain why subtracting \(k^2\) from both sides was useful in simplifying this problem?
4. How would the solution change if we had \((k - 2)(k + 6) < k^2 + 24\) instead?
5. What is the graphical interpretation of this inequality on a number line?

**Tip:** When solving inequalities, carefully check for any changes in inequality direction when dividing by negative numbers.

Solve the following inequality: (k - 2)(k + 6) ≤ k^2 + 24

Learn how to solve the quadratic inequality (k - 2)(k + 6) ≤ k^2 + 24 with a step-by-step solution. This problem explores algebraic concepts including quadratic expansion and inequality solving techniques. Suitable for high school students in Grades 10-12.

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