Let's find the resulting function, [tex]\( (f+g)(x) \)[/tex], by adding the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] step-by-step.
Given:
[tex]\[
f(x) = 4x^2 + 7x - 3
\][/tex]
[tex]\[
g(x) = 6x^3 - 7x^2 - 5
\][/tex]
To find [tex]\( (f+g)(x) \)[/tex], we add [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
(f+g)(x) = f(x) + g(x)
\][/tex]
[tex]\[
(f+g)(x) = (4x^2 + 7x - 3) + (6x^3 - 7x^2 - 5)
\][/tex]
Now, let's combine like terms:
1. [tex]\( x^3 \)[/tex] term:
Only [tex]\( g(x) \)[/tex] has a [tex]\( x^3 \)[/tex] term:
[tex]\[
6x^3
\][/tex]
2. [tex]\( x^2 \)[/tex] term:
Combine the [tex]\( x^2 \)[/tex] terms from [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
4x^2 - 7x^2 = -3x^2
\][/tex]
3. [tex]\( x \)[/tex] term:
Only [tex]\( f(x) \)[/tex] has an [tex]\( x \)[/tex] term:
[tex]\[
7x
\][/tex]
4. Constant term:
Combine the constants from [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
-3 - 5 = -8
\][/tex]
Now, combine all these terms together:
[tex]\[
(f+g)(x) = 6x^3 - 3x^2 + 7x - 8
\][/tex]
Thus, the resulting function is:
[tex]\[
(f+g)(x) = 6x^3 - 3x^2 + 7x - 8
\][/tex]
From the given options, this matches Option D.
Therefore, the correct answer is:
[tex]\[
\boxed{\text{D. } (f+g)(x)=6 x^3-3 x^2+7 x-8}
\][/tex]
