Let's solve this problem part by part.
### Part A: Amount of canned food collected so far
First, we need to add the number of cans collected by each friend.
- Jessa collected: [tex]\(7xy + 3\)[/tex]
- Tyree collected: [tex]\(3x^2 - 4\)[/tex]
- Ben collected: [tex]\(5x^2\)[/tex]
To find the total amount collected so far, we combine these expressions:
[tex]\[ 7xy + 3 + 3x^2 - 4 + 5x^2 \][/tex]
Combine like terms:
- [tex]\(3x^2 + 5x^2 = 8x^2\)[/tex]
- Constants: [tex]\(3 - 4 = -1\)[/tex]
So, the total collected so far is:
[tex]\[ 8x^2 + 7xy - 1 \][/tex]
Therefore, the expression for the amount of canned food collected so far by the three friends is:
[tex]\[ 8x^2 + 7xy - 1 \][/tex]
### Part B: Number of cans still needed to meet the goal
The goal for the collection is given by the expression:
[tex]\[ 10x^2 - 4xy + 12 \][/tex]
We need to find how many more cans are needed by subtracting the collected amount from the goal:
[tex]\[ \text{Goal} - \text{Collected so far} = 10x^2 - 4xy + 12 - (8x^2 + 7xy - 1) \][/tex]
Now, distribute the negative sign across the collected amount expression:
[tex]\[ 10x^2 - 4xy + 12 - 8x^2 - 7xy + 1 \][/tex]
Combine like terms:
- For [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 - 8x^2 = 2x^2\)[/tex]
- For [tex]\(xy\)[/tex] terms: [tex]\(-4xy - 7xy = -11xy\)[/tex]
- Constants: [tex]\(12 + 1 = 13\)[/tex]
So, the expression representing the number of cans the friends still need to collect to meet their goal is:
[tex]\[ 2x^2 - 11xy + 13 \][/tex]
In summary:
- Part A: The total collection so far is [tex]\(8x^2 + 7xy - 1\)[/tex].
- Part B: The number of cans still needed is [tex]\(2x^2 - 11xy + 13\)[/tex].
