Question

vanessa uses the expressions \\((3x^2 + 5x + 10)\\) and \\((x^2 - 3x - 1)\\) to represent the length and width of her patio. which expression represents the area (\\(lw\\)) of vanessas patio?

\\(3x^4 - 4x^3 - 8x^2 - 35x - 10\\)
\\(3x^4 + 14x^3 + 22x^2 + 25x - 10\\)
\\(3x^4 - 4x^3 + 22x^2 + 35x + 10\\)
\\(3x^4 + 14x^3 + 28x^2 + 35x + 10\\)

Explanation:

Set up the area formula

The area of a rectangular patio is given by the product of its length and width:

$$ \text{Area} = l \times w $$

Given:

$$ l = 3x^2 + 5x + 10 $$

$$ w = x^2 - 3x - 1 $$

Therefore, the area expression is:

$$ \text{Area} = (3x^2 + 5x + 10)(x^2 - 3x - 1) $$

Multiply the polynomials

Using the Polynomial Multiplication knowledge point

$$
LATEXBLOCK0
$$

Combine like terms

Using the Polynomial Multiplication knowledge point

$$
LATEXBLOCK1
$$

Answer:

• (A) \(3x^4 - 4x^3 - 8x^2 - 35x - 10\) (Correct answer)
• (B) \(3x^4 + 14x^3 + 22x^2 + 25x - 10\)
• (C) \(3x^4 - 4x^3 + 22x^2 + 35x + 10\)
• (D) \(3x^4 + 14x^3 + 28x^2 + 35x + 10\)