To find the amount of money the friends still need to collect, we should first determine the total amount they have already collected and then subtract this from their goal.
1. The goal amount is given by:
[tex]\[ 9x^2 - 4xy + 6 \][/tex]
2. The amounts collected by each of the friends are:
- Martin: [tex]\[ 2xy + 12 \][/tex]
- Tyreese: [tex]\[ x^2 \][/tex]
- Braydon: [tex]\[ 3x^2 - 5 \][/tex]
3. Add the amounts collected by Martin, Tyreese, and Braydon to find the total collected amount:
[tex]\[ (2xy + 12) + (x^2) + (3x^2 - 5) \][/tex]
4. Combine like terms in the expression:
[tex]\[ = 2xy + 12 + x^2 + 3x^2 - 5 \][/tex]
[tex]\[ = 2xy + 4x^2 + 7 \][/tex]
5. Subtract this total collected amount from the goal amount to find the amount needed:
[tex]\[ (9x^2 - 4xy + 6) - (2xy + 4x^2 + 7) \][/tex]
[tex]\[ = 9x^2 - 4xy + 6 - 2xy - 4x^2 - 7 \][/tex]
6. Combine like terms in this subtraction to get the final expression:
[tex]\[ = 9x^2 - 4x^2 - 4xy - 2xy + 6 - 7 \][/tex]
[tex]\[ = 5x^2 - 6xy - 1 \][/tex]
Therefore, the amount of money the friends still need to collect to meet their goal is:
[tex]\[ 5x^2 - 6xy - 1 \][/tex]
This corresponds to the answer choice:
[tex]\[ \boxed{5x^2 - 6xy - 1} \][/tex]
