The expression to be expanded and simplified is:

$(x^2 + 3x + 2)(x - 8)$

Let's expand this step by step:

Step 1: Apply the distributive property

We will distribute each term in the first polynomial $(x^2 + 3x + 2)$ to every term in the second polynomial $(x - 8)$:

$(x^2 + 3x + 2)(x - 8) = x^2(x - 8) + 3x(x - 8) + 2(x - 8)$

Step 2: Distribute each term

1. $x^2(x - 8) = x^3 - 8x^2$
2. $3x(x - 8) = 3x^2 - 24x$
3. $2(x - 8) = 2x - 16$

Step 3: Combine all terms

Now, we combine the results from Step 2:

$x^3 - 8x^2 + 3x^2 - 24x + 2x - 16$

Step 4: Simplify by combining like terms

• Combine the $x^2$ terms: $-8x^2 + 3x^2 = -5x^2$
• Combine the $x$ terms: $-24x + 2x = -22x$

Thus, the fully simplified expression is:

$x^3 - 5x^2 - 22x - 16$

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

Here are 5 related questions to explore:

1. How can the distributive property be used for more complex polynomials?
2. What are the key steps to simplifying polynomial expressions?
3. How do you determine which terms to combine when simplifying?
4. How can polynomial multiplication be applied in real-world problems?
5. How would you factor the resulting polynomial?

Tip: Always double-check for any like terms after expanding to ensure the expression is fully simplified.