Vanessa uses the expression (3x^2+5x+10) and (x^2-3x-1) to represent the length and width of her patio. which expression represents the area (iw) of vanessa's patio?

Since you know the length and the width of Vanessa's patio, (3x^2+5x+10) and 
(x^2-3x-1), to find the expression that represents its area you would multiply the length and width. This would be represented by this expression:
(3x^2+5x+10)(x^2-3x-1)

When you multiply the two expressions, you must distribute each term to one another. This would look like this:
(3x^2*x^2)+(3x^2*-3x)+(3x^2*-1)+(5x*x^2)+(5x*-3x)+(5x*-1)+(10*x^2)+(10*-3x)+(10*-1)

When simplified, you should get:
3x^4-9x^3-3x^2+5x^3-15x^2-5x+10x^2-30x-10

Then, combine your like terms:
3x^4-4x^3-8x^2-35x-10

Your final answer would be that the expression representing the area of Vanessa's patio is 3x^4-4x^3-8x^2-35x-10