The area of rectangle with length l and width w is
[tex]A=l\cdot w.[/tex]
If the length of rectangle is expressed as [tex]l=3x^2 + 5x + 10[/tex] and the width of rectangle is expressed as [tex]w=x^2-3x-1[/tex], then the area of rectangle is
[tex]A=(3x^2 + 5x + 10)(x^2-3x-1)=3x^4-4x^3-8x^2-35x-10.[/tex]
Area of a rectangle is the product of its dimensions. The area of Vanessa's patio with specified length and width is [tex]Area = 2x^4 - 4x^3 -8x^2 -35x - 10[/tex] squared units.
A patio is usually rectangular. If its length and width is given, then we can use their product as measure of its area as the area of a rectangle is the product of its length and width.
Since we're given that
Thus, the area is calculated as
[tex]\begin{aligned} Area &= (3x^2 + 5x + 10) \times (x^2 - 3x - 1)\\&= 3x^2 (x^2 - 3x - 1) + 5x (x^2 - 3x - 1) + 10 (x^2 - 3x - 1)\\&= 2x^{2 + 2} - 9x^{2 + 1} - 3x^2 + 5x^{1 + 2} - 15x^{1 + 1} - 5x + 10x^2 - 30x - 10\\&= 2x^4 - 9x^3 - 3x^2 + 5x^3 - 15x^2 - 5x + 10x^2 - 30x - 10\\&= 2x^4 - 4x^3 -8x^2 -35x - 10\\\end{aligned}[/tex]
(We used three properties. One is distributive property of multiplication over addition and of distributive property of addition over multiplication and addition of coefficients of like terms' addition)
Thus,
The area of Vanessa's patio with specified length and width is given as
[tex]Area = 2x^4 - 4x^3 -8x^2 -35x - 10[/tex] squared units.
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