To solve the problem [tex]\(\left(9 a^2 - 7 b + 3 c^3 - 4\right) - \left(6 a^2 + 6 b - 3 c^3 - 4\right)\)[/tex], follow these detailed steps:
1. Rewrite the expression to distribute the negative sign:
[tex]\[
(9 a^2 - 7 b + 3 c^3 - 4) - (6 a^2 + 6 b - 3 c^3 - 4)
\][/tex]
This becomes:
[tex]\[
9 a^2 - 7 b + 3 c^3 - 4 - 6 a^2 - 6 b + 3 c^3 + 4
\][/tex]
2. Group the like terms together:
[tex]\[
(9 a^2 - 6 a^2) + (-7 b - 6 b) + (3 c^3 + 3 c^3) + (-4 + 4)
\][/tex]
3. Combine the like terms:
- For [tex]\(a^2\)[/tex] terms: [tex]\(9 a^2 - 6 a^2 = 3 a^2\)[/tex]
- For [tex]\(b\)[/tex] terms: [tex]\(-7 b - 6 b = -13 b\)[/tex]
- For [tex]\(c^3\)[/tex] terms: [tex]\(3 c^3 + 3 c^3 = 6 c^3\)[/tex]
- For constant terms: [tex]\(-4 + 4 = 0\)[/tex]
4. Write the final simplified expression:
[tex]\[
3 a^2 - 13 b + 6 c^3
\][/tex]
Thus, the simplified expression is [tex]\(\boxed{3 a^2 - 13 b + 6 c^3}\)[/tex].
