Certainly! Let's solve the problem step-by-step.
### Part A: Total Canned Food Collected So Far
To determine the total number of canned food items collected by all three friends, we need to sum up the canned food collected by Jessa, Tyree, and Ben.
1. Jessa's Contribution:
[tex]\[
3x^2
\][/tex]
2. Tyree's Contribution:
[tex]\[
5x^2 - 8
\][/tex]
3. Ben's Contribution:
[tex]\[
2xy + 4
\][/tex]
Now, we add these expressions together:
[tex]\[
3x^2 + (5x^2 - 8) + (2xy + 4)
\][/tex]
Combine like terms:
[tex]\[
3x^2 + 5x^2 + 2xy - 8 + 4
\][/tex]
[tex]\[
(3x^2 + 5x^2) + 2xy + (-8 + 4)
\][/tex]
[tex]\[
8x^2 + 2xy - 4
\][/tex]
So, the total number of cans collected so far is:
[tex]\[
8x^2 + 2xy - 4
\][/tex]
### Part B: Cans Still Needed to Meet the Goal
The goal for the total number of canned food items is represented by the expression:
[tex]\[
12x^2 - 2xy + 3
\][/tex]
Now, we need to find out how many more cans need to be collected. This can be done by subtracting the amount collected so far (from Part A) from the goal:
[tex]\[
(12x^2 - 2xy + 3) - (8x^2 + 2xy - 4)
\][/tex]
Distribute the negative sign through the second expression:
[tex]\[
12x^2 - 2xy + 3 - 8x^2 - 2xy + 4
\][/tex]
Combine like terms:
[tex]\[
(12x^2 - 8x^2) + (-2xy - 2xy) + (3 + 4)
\][/tex]
[tex]\[
4x^2 - 4xy + 7
\][/tex]
So, the number of cans still needed to meet their goal is:
[tex]\[
4x^2 - 4xy + 7
\][/tex]
Putting it all together, we have:
### Final Answer
Part A: The total amount of canned food collected so far by the three friends is:
[tex]\[
8x^2 + 2xy - 4
\][/tex]
Part B: The number of cans the friends still need to collect to meet their goal is:
[tex]\[
4x^2 - 4xy + 7
\][/tex]
