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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area of Vanessa's patio = length × width
Given, length = (3x² + 5x + 10)
width = (x² - 3x - 1)
area of Vanessa's patio = (3x² + 5x + 10)(x² - 3x - 1)
= 3x²(x² - 3x - 1) + 5x(x² - 3x - 1) + 10(x² - 3x - 1)
= 3x⁴ - 9x³ - 3x² + 5x³ - 15x² - 5x + 10x² - 30x - 10
= 3x⁴ - 4x³ - 8x² - 35x - 10
hence, expression of Vanessa's patio = (3x⁴ - 4x³ - 8x² - 35x - 10)
Given, length = (3x² + 5x + 10)
width = (x² - 3x - 1)
area of Vanessa's patio = (3x² + 5x + 10)(x² - 3x - 1)
= 3x²(x² - 3x - 1) + 5x(x² - 3x - 1) + 10(x² - 3x - 1)
= 3x⁴ - 9x³ - 3x² + 5x³ - 15x² - 5x + 10x² - 30x - 10
= 3x⁴ - 4x³ - 8x² - 35x - 10
hence, expression of Vanessa's patio = (3x⁴ - 4x³ - 8x² - 35x - 10)
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