Certainly! Let's simplify the given expression step by step. We start with:
[tex]\[
\left(3x^2 - 3 + 9x^3\right) - \left(4x^3 - 2x^2 + 16\right)
\][/tex]
First, we distribute the negative sign across the terms in the second polynomial:
[tex]\[
= 3x^2 - 3 + 9x^3 - 4x^3 + 2x^2 - 16
\][/tex]
Next, we combine like terms. We start by combining the [tex]\(x^3\)[/tex] terms:
[tex]\[
9x^3 - 4x^3 = 5x^3
\][/tex]
Then, we combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
3x^2 + 2x^2 = 5x^2
\][/tex]
Finally, we combine the constant terms:
[tex]\[
-3 - 16 = -19
\][/tex]
Putting all the simplified terms together, we get:
[tex]\[
5x^3 + 5x^2 - 19
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{5x^3 + 5x^2 - 19}
\][/tex]
