To find [tex]\((f+g)(x)\)[/tex], we need to add the functions [tex]\(f(x) = 3x^2 + 6x - 5\)[/tex] and [tex]\(g(x) = 4x^3 - 5x^2 + 6\)[/tex].
First, let's write down the given functions:
[tex]\[
f(x) = 3x^2 + 6x - 5
\][/tex]
[tex]\[
g(x) = 4x^3 - 5x^2 + 6
\][/tex]
To find [tex]\((f+g)(x)\)[/tex], we simply add [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f+g)(x) = f(x) + g(x)
\][/tex]
Substituting the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we get:
[tex]\[
(f+g)(x) = (3x^2 + 6x - 5) + (4x^3 - 5x^2 + 6)
\][/tex]
Now, let's combine like terms:
[tex]\[
(f+g)(x) = 4x^3 + 3x^2 + 6x + 1
\][/tex]
So, the expression for [tex]\((f+g)(x)\)[/tex] is:
[tex]\[
(f+g)(x) = 4x^3 + 3x^2 + 11x + 1
\][/tex]
Looking at the provided options:
A. [tex]\((-4x^3 + 8x^2 + 6x - 11)\)[/tex]
B. [tex]\((7x^3 + x^2 + 1)\)[/tex]
C. [tex]\((4x^3 - 2x^2 + 6x + 1)\)[/tex]
D. [tex]\((4x^3 + 3x^2 + 11x + 1)\)[/tex]
The correct option is:
D. [tex]\((4x^3 + 3x^2 + 11x + 1)\)[/tex]
Therefore, the correct answer is option D.
