To find the expression \((f+g)(x)\), we need to add the given functions \(f(x)\) and \(g(x)\) together. Let's break this down step by step.
Given:
[tex]\[ f(x) = 4x^2 + 7x - 3 \][/tex]
[tex]\[ g(x) = 6x^3 - 7x^2 - 5 \][/tex]
We need to add these functions together term by term.
1. First, identify the powers of \(x\) in each function and their coefficients:
For \(f(x)\):
[tex]\[ f(x) = 4x^2 + 7x - 3 \][/tex]
- Coefficient of \(x^2\): \(4\)
- Coefficient of \(x\): \(7\)
- Constant term: \(-3\)
For \(g(x)\):
[tex]\[ g(x) = 6x^3 - 7x^2 - 5 \][/tex]
- Coefficient of \(x^3\): \(6\)
- Coefficient of \(x^2\): \(-7\)
- Constant term: \(-5\)
2. Next, add the coefficients of terms with the same powers of \(x\):
- For \(x^3\):
[tex]\[ \text{Coefficient of } x^3 = 6 \][/tex]
- For \(x^2\):
[tex]\[ \text{Coefficient of } x^2 = 4 - 7 = -3 \][/tex]
- For \(x\):
[tex]\[ \text{Coefficient of } x = 7 \][/tex]
- For the constant term:
[tex]\[ \text{Constant term} = -3 - 5 = -8 \][/tex]
3. Combine these results to write \((f+g)(x)\):
So, \((f+g)(x)\) is:
[tex]\[ (f+g)(x) = 6x^3 - 3x^2 + 7x - 8 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{6x^3 - 3x^2 + 7x - 8} \][/tex]
Comparing our result with the given options, we see that it matches option A:
[tex]\[ (f+g)(x)=6 x^3-3 x^2+7 x-8 \][/tex]
Therefore, the correct choice is:
A. [tex]\((f+g)(x)=6 x^3-3 x^2+7 x-8\)[/tex]
