To subtract the expressions [tex]\((9a^2 - 7b + 3c^3 - 4)\)[/tex] and [tex]\((6a^2 + 6b - 3c^3 - 4)\)[/tex], follow these steps:
1. Align the like terms: Place the two expressions in such a way that like terms are aligned for easy subtraction.
Original expressions:
[tex]\[
(9a^2 - 7b + 3c^3 - 4) - (6a^2 + 6b - 3c^3 - 4)
\][/tex]
2. Distribute the negative sign: Distribute the negative sign across the second set of parentheses:
[tex]\[
9a^2 - 7b + 3c^3 - 4 - 6a^2 - 6b + 3c^3 + 4
\][/tex]
3. Combine like terms:
- Coefficients of [tex]\(a^2\)[/tex]:
[tex]\[
9a^2 - 6a^2 = 3a^2
\][/tex]
- Coefficients of [tex]\(b\)[/tex]:
[tex]\[
-7b - 6b = -13b
\][/tex]
- Coefficients of [tex]\(c^3\)[/tex]:
[tex]\[
3c^3 + 3c^3 = 6c^3
\][/tex]
- Constants:
[tex]\[
-4 + 4 = 0
\][/tex]
4. Write the resulting expression:
[tex]\[
3a^2 - 13b + 6c^3
\][/tex]
Therefore, the simplified expression after subtracting [tex]\((6a^2 + 6b - 3c^3 - 4)\)[/tex] from [tex]\((9a^2 - 7b + 3c^3 - 4)\)[/tex] is:
[tex]\[
3a^2 - 13b + 6c^3
\][/tex]
So, the correct choice is:
[tex]\[
\boxed{3a^2 - 13b + 6c^3}
\][/tex]
