To decorate for a party, rob and loretta were each given a bag of balloons. there were 40 balloons between both bags. rob used 8 balloons from his bag, and loretta used 8 balloons from hers. the product of the number of balloons left in each bag is no more than 44. let x represent the number of balloons that rob had in his bag before he used any. which choice represents the number of balloons left in their bags? a.) x^2-40x-300< 0 b.) x^2-40x=300 c.) x^2-40x-300=0 d.) x^2-40x+300> =0 if rob originally had more balloons than loretta, what is the least number of balloons that rob could have had before using any? a.) 30 b.) 20 c.) 10

Answers:
Part (a): x² - 40x + 300 ≥ 0
Part (b): 30 balloons

Explanation:
Part (a):
We are given that:
Sum of balloons in both bags is 40
Number of balloons in Rob's bag is x
This means that number of balloons in Loretta's bag is 40-x

Now, we know that each of them used 8 balloons from his/her bag.
This means that:
Number of balloons left in Rob's is x-8 balloons
Number of balloons left is Loretta's bag is (40-x)-8 = 32-x balloons

Finally, we know that the product of the balloons left in both bags is no more than 44. This means that the product could be 44 or less.
Therefore:
(x-8)(32-x) ≤ 44
x(32) + x(-x) - 8(32) - 8(-x) ≤ 44
32x - x² -256 + 8x - 44 ≤ 0
-x² + 40x - 300 ≤ 0

Now, multiply both sides by -1. Remember that when multiplying by negative number, the inequality sign is reversed:
-1(-x² + 40x - 300) ≥ 0
x² - 40x + 300 ≥ 0

Part (b):
Before using any balloons:
Rob had x balloons
Loretta had 40-x balloons
We are given that originally Rob had more balloons than Loretta
Let's check examine the given choices:
1- If Rob had 40 balloons:
Loretta had = 40 - 30 = 10 balloons
In this case, Rob originally had more balloons than Loretta > accepted answer
2- If Rob had 20 balloons:
Loretta had = 40 - 20 = 20 balloons
In this case, They both originally had equal number of balloons
> rejected answer
3- If Rob had 10 balloons:
Loretta had = 40 - 10 = 30 balloons
In this case, Loretta originally had more balloons than Rob
> rejected answer

Hope this helps :)