To solve this problem, let's find the area of a rectangle with height $x + 4$ and width $x^2 + 3x + 2$. The area can be found by multiplying the height and width:

1. Set up the multiplication: $(x + 4)(x^2 + 3x + 2)$

2. Distribute $x$ across each term in the width polynomial:

   • $x \cdot x^2 = x^3$
   • $x \cdot 3x = 3x^2$
   • $x \cdot 2 = 2x$

   So, multiplying $x$ gives: $x^3 + 3x^2 + 2x$.

3. Distribute $4$ across each term in the width polynomial:

   • $4 \cdot x^2 = 4x^2$
   • $4 \cdot 3x = 12x$
   • $4 \cdot 2 = 8$

   So, multiplying $4$ gives: $4x^2 + 12x + 8$.

4. Combine all terms: Now we add the two results from the distribution step: $x^3 + 3x^2 + 2x + 4x^2 + 12x + 8$

5. Combine like terms:

   • $x^3$ term: $x^3$
   • $x^2$ terms: $3x^2 + 4x^2 = 7x^2$
   • $x$ terms: $2x + 12x = 14x$
   • Constant term: $8$

   So, the area of the rectangle is: $x^3 + 7x^2 + 14x + 8$

This is the polynomial that represents the area of the rectangle.

Would you like more details on any specific step, or do you have any other questions?

Here are some related questions to deepen your understanding:

1. How would you find the area if the height or width had additional terms?
2. How can you verify that your polynomial expression for the area is correct?
3. What would change if the height was $x + 5$ instead of $x + 4$?
4. How would you approach this problem if asked to factorize the polynomial instead of expanding it?
5. Can you identify other situations where the distributive property helps in algebra?

Tip: When multiplying polynomials, keeping terms organized can make combining like terms much easier.