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Let's solve the given inequality:

\[
(p - 2)(p + 4) \leq p^2 + 16
\]

### Step 1: Expand the left-hand side
\[
(p - 2)(p + 4) = p^2 + 4p - 2p - 8 = p^2 + 2p - 8
\]

So the inequality becomes:

\[
p^2 + 2p - 8 \leq p^2 + 16
\]

### Step 2: Simplify the inequality
Subtract \(p^2\) from both sides:

\[
2p - 8 \leq 16
\]

### Step 3: Solve for \(p\)
Add 8 to both sides:

\[
2p \leq 24
\]

Now divide both sides by 2:

\[
p \leq 12
\]

### Step 4: Check for extraneous solutions
Since the original inequality involves a product \((p - 2)(p + 4)\), it’s essential to analyze any restrictions from the quadratic factors. However, no restrictions directly affect the inequality beyond the solution found.

Thus, the solution to the inequality is:

\[
p \leq 12
\]

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Do you have any follow-up questions or need a more detailed explanation?  
Here are five related questions you might find interesting:
1. What happens if we change the inequality sign to \(>\)?
2. How do we solve inequalities involving absolute values?
3. What are the critical points of a quadratic function, and how do they relate to inequalities?
4. How do quadratic inequalities differ from linear inequalities?
5. Can we solve similar inequalities graphically?

**Tip:** Always expand and simplify carefully to avoid missing small algebraic errors!

Learn how to solve the inequality (p - 2)(p + 4) ≤ p² + 16 step-by-step, covering key algebraic concepts such as expansion, simplification, and solving linear inequalities. This is suitable for students in Grades 8-10.

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