To find [tex]\((f+g)(x)\)[/tex], we need to add the polynomials [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together. Let's break this down step by step:
1. Given the functions:
[tex]\( f(x) = 4x^2 + 3x - 2 \)[/tex]
[tex]\( g(x) = 6x^3 - 3x^2 - 4 \)[/tex]
2. We need to find [tex]\( (f + g)(x) \)[/tex].
3. Add [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
(f + g)(x) = f(x) + g(x) = (4x^2 + 3x - 2) + (6x^3 - 3x^2 - 4)
\][/tex]
4. Combine like terms:
[tex]\[
= 6x^3 + 4x^2 - 3x^2 + 3x - 2 - 4
\][/tex]
5. Simplify the combined polynomial:
[tex]\[
= 6x^3 + (4x^2 - 3x^2) + 3x + (-2 - 4)
\][/tex]
[tex]\[
= 6x^3 + x^2 + 3x - 6
\][/tex]
Therefore, we find that:
[tex]\[
(f + g)(x) = 6x^3 + x^2 + 3x - 6
\][/tex]
Looking at the provided options:
A. [tex]\(10x^3 - 6\)[/tex]
B. [tex]\(-6x^3 + 7x^2 + 3x + 2\)[/tex]
C. [tex]\(6x^3 + 4x^2 - 6\)[/tex]
D. [tex]\(6x^3 + x^2 + 3x - 6\)[/tex]
The correct answer is:
[tex]\[
\boxed{D. 6x^3 + x^2 + 3x - 6}
\][/tex]
