To solve this problem, we'll break it down into two parts: finding the total number of canned foods collected by the three friends and finding out how many more are needed to meet their goal.
### Part A: Expression for the Amount of Canned Food Collected So Far
Let's write down the number of cans each friend has collected:
- Jessa: [tex]\( 3x^2 \)[/tex]
- Tyree: [tex]\( 5x^2 - 8 \)[/tex]
- Ben: [tex]\( 3xy + 4 \)[/tex]
To find the total number of cans collected so far, we need to add these expressions together:
[tex]\[
\text{Total collected} = (3x^2) + (5x^2 - 8) + (3xy + 4)
\][/tex]
Next, combine like terms:
[tex]\[
3x^2 + 5x^2 + 3xy - 8 + 4
\][/tex]
Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
(3x^2 + 5x^2) = 8x^2
\][/tex]
Combine the constant terms:
[tex]\[
-8 + 4 = -4
\][/tex]
Putting it all together:
[tex]\[
8x^2 + 3xy - 4
\][/tex]
So, the expression representing the total amount of canned food collected so far by the three friends is:
[tex]\[
8x^2 + 3xy - 4
\][/tex]
### Part B: Expression for the Number of Cans Needed to Meet Their Goal
The goal for the canned food collection is given by:
[tex]\[
12x^2 - 2xy + 3
\][/tex]
To find how many more cans are needed, we subtract the total collected amount from the goal amount:
[tex]\[
\text{Remaining to collect} = (\text{Goal}) - (\text{Total collected})
\][/tex]
Substitute the expressions we found:
[tex]\[
\text{Remaining to collect} = (12x^2 - 2xy + 3) - (8x^2 + 3xy - 4)
\][/tex]
Distribute the subtraction:
[tex]\[
12x^2 - 2xy + 3 - 8x^2 - 3xy + 4
\][/tex]
Combine like terms:
[tex]\[
(12x^2 - 8x^2) + (-2xy - 3xy) + (3 + 4)
\][/tex]
Simplify:
[tex]\[
4x^2 - 5xy + 7
\][/tex]
So, the expression representing the number of cans the friends still need to collect to meet their goal is:
[tex]\[
4x^2 - 5xy + 7
\][/tex]
### Summary
Part A:
The expression for the total amount of canned food collected so far is:
[tex]\[
8x^2 + 3xy - 4
\][/tex]
Part B:
The expression for the number of cans still needed to meet their goal is:
[tex]\[
4x^2 - 5xy + 7
\][/tex]
