To find [tex]\((f+g)(x)\)[/tex], we need to add the two given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given the functions:
[tex]\[ f(x) = 4x^2 + 5x - 3 \][/tex]
[tex]\[ g(x) = 4x^3 - 3x^2 + 5 \][/tex]
We want to find the sum [tex]\( (f + g)(x) \)[/tex], which is:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Let's add the two functions step by step:
1. Start with the polynomial terms involving [tex]\( x^3 \)[/tex]:
[tex]\[ \text{From } g(x): 4x^3 \][/tex]
So for [tex]\(x^3\)[/tex], we have:
[tex]\[ 4x^3 \][/tex]
2. Next, combine the polynomial terms involving [tex]\( x^2 \)[/tex]:
[tex]\[ \text{From } f(x): 4x^2 \][/tex]
[tex]\[ \text{From } g(x): -3x^2 \][/tex]
Adding these terms together:
[tex]\[ 4x^2 - 3x^2 = x^2 \][/tex]
3. Now, combine the terms involving [tex]\( x \)[/tex]:
[tex]\[ \text{From } f(x): 5x \][/tex]
[tex]\[ \text{There is no } x \text{ term in } g(x) \][/tex]
So the term involving [tex]\( x \)[/tex] is:
[tex]\[ 5x \][/tex]
4. Lastly, combine the constant terms:
[tex]\[ \text{From } f(x): -3 \][/tex]
[tex]\[ \text{From } g(x): 5 \][/tex]
Adding the constants:
[tex]\[ -3 + 5 = 2 \][/tex]
Putting it all together, we get:
[tex]\[ (f + g)(x) = 4x^3 + x^2 + 5x + 2 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D. \; (f+g)(x) = 4x^3 + x^2 + 5x + 2} \][/tex]
