Vanessa uses the expressions (3x2 + 5x + 10) and (x2 – 3x – 1) to represent the length and width of her patio. Which expression represents the area (lw) of Vanessa’s patio?
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To find the expression that represents the area of Vanessa's patio, you need to multiply the two expressions that represent the length and width.

Given:

- Length: \( 3x^2 + 5x + 10 \)
- Width: \( x^2 - 3x - 1 \)

The area \( A \) is calculated by multiplying these two expressions:

\[
A = (3x^2 + 5x + 10)(x^2 - 3x - 1)
\]

Let's expand the multiplication:

\[
A = 3x^2 \cdot (x^2 - 3x - 1) + 5x \cdot (x^2 - 3x - 1) + 10 \cdot (x^2 - 3x - 1)
\]

First, multiply each term in the length expression by each term in the width expression:

1. \( 3x^2 \cdot x^2 = 3x^4 \)
2. \( 3x^2 \cdot (-3x) = -9x^3 \)
3. \( 3x^2 \cdot (-1) = -3x^2 \)
4. \( 5x \cdot x^2 = 5x^3 \)
5. \( 5x \cdot (-3x) = -15x^2 \)
6. \( 5x \cdot (-1) = -5x \)
7. \( 10 \cdot x^2 = 10x^2 \)
8. \( 10 \cdot (-3x) = -30x \)
9. \( 10 \cdot (-1) = -10 \)

Now, add all these terms together:

\[
A = 3x^4 - 9x^3 - 3x^2 + 5x^3 - 15x^2 - 5x + 10x^2 - 30x - 10
\]

Combine like terms:

\[
A = 3x^4 + (-9x^3 + 5x^3) + (-3x^2 - 15x^2 + 10x^2) + (-5x - 30x) - 10
\]

\[
A = 3x^4 - 4x^3 - 8x^2 - 35x - 10
\]

Thus, the expression that represents the area of Vanessa's patio is:

\[
\boxed{3x^4 - 4x^3 - 8x^2 - 35x - 10}
\]

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How do you expand polynomials?
2. What are like terms in polynomial expressions?
3. How can you check the correctness of polynomial multiplication?
4. What is the significance of each term in a polynomial?
5. How do you simplify polynomial expressions?
6. Can you factor the resulting polynomial expression?
7. What is the degree of a polynomial, and how is it determined?
8. How would the area change if the length or width was modified?

**Tip:** When multiplying polynomials, always ensure that you distribute each term in the first polynomial to every term in the second polynomial to get the correct result.

Learn how to find the area of Vanessa's patio using polynomial multiplication. Follow a step-by-step explanation of multiplying polynomials, combining like terms, and deriving the area expression. Suitable for high school students studying algebra.

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