Find each product.

{ this means the square root sign.

1. (3+2{20)(5+3{45)

2. The length and width of the rectangle are x-5 and 2x-6

a. Write a trinomial that represents that area of the rectangle.

b. What is the smallest integer value of x that will give positive values for the length and the width of the rectangle?

just answered that in

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1. (3+2sqrt 20 )(5+3sqrt 45) ???

use FOIL
15 + 9 sqrt(900) + 10 sqrt(900) + 900
but sqrt 900 = 30
so 15 + 19*30 + 900

2. The length and width of the rectangle are x-5 and 2x-6

a. Write a trinomial that represents that area of the rectangle.
(x-5)(2x-6) = 2 x^2 -16 x + 30

b. What is the smallest integer value of x that will give positive values for the length and the width of the rectangle?
If x is <5, length is negative
if x < 3, width is negative
so 3

I multiplied wrong, when I foiled.

To find each product, we will simplify the given expressions step by step.

1. (3+2{20)(5+3{45)
First, let's simplify the square roots by finding the values inside them.
{20 = 2 * {5 = 2 * √5
{45 = 3 * {5 = 3 * √5

Now, we can substitute these values back into the original expression:
(3 + 2 * 2 * √5)(5 + 3 * 3 * √5)

Next, we will multiply the terms inside each parentheses using the distributive property:
(3 + 4√5)(5 + 9√5)

Now, let's multiply each term in the first parentheses by each term in the second parentheses:
3*5 + 3*9√5 + 4√5*5 + 4√5*9√5

Simplifying each term:
15 + 27√5 + 20√5 + 36 * 5

Combining like terms:
15 + 47√5 + 180

The final product is 195 + 47√5.

2. The length and width of the rectangle are given as x-5 and 2x-6, respectively.

a. To find the trinomial that represents the area of the rectangle, we multiply the length and width:
Area = (x-5) * (2x-6)

Now, we can apply the distributive property to expand the expression:
Area = 2x^2 - 6x - 10x + 30

Simplifying the expression:
Area = 2x^2 - 16x + 30

The trinomial that represents the area of the rectangle is 2x^2 - 16x + 30.

b. We want to find the smallest integer value of x that will give positive values for the length and width of the rectangle.
Since length and width cannot be negative, we set the quadratic trinomial greater than zero and solve for x.

2x^2 - 16x + 30 > 0

To solve this inequality, we can factor the quadratic:
(2x - 10)(x - 3) > 0

Setting each factor equal to zero, we get:
2x - 10 = 0 => 2x = 10 => x = 5
x - 3 = 0 => x = 3

Now, we can use a sign chart or test intervals to determine the values of x that make the inequality true:
For x < 3, (2x - 10) and (x - 3) will be negative, making the product positive.
For x between 3 and 5, (2x - 10) will be positive and (x - 3) will be negative, making the product negative.
For x > 5, (2x - 10) and (x - 3) will both be positive, making the product positive.

Therefore, the smallest integer value of x that will give positive values for the length and the width of the rectangle is x = 5.