To find the sum of the given polynomials [tex]\((4x^3 - 2x - 9)\)[/tex] and [tex]\((2x^3 + 5x + 3)\)[/tex], we will add the corresponding coefficients of the same powers of [tex]\(x\)[/tex]. Here’s how you can do it step-by-step:
1. Identify and list the coefficients of each power of [tex]\(x\)[/tex]:
For the first polynomial [tex]\((4x^3 - 2x - 9)\)[/tex]:
- [tex]\(x^3\)[/tex] term: [tex]\(4\)[/tex]
- [tex]\(x^2\)[/tex] term: [tex]\(0\)[/tex] (since there is no [tex]\(x^2\)[/tex] term, the coefficient is [tex]\(0\)[/tex])
- [tex]\(x\)[/tex] term: [tex]\(-2\)[/tex]
- Constant term: [tex]\(-9\)[/tex]
For the second polynomial [tex]\((2x^3 + 5x + 3)\)[/tex]:
- [tex]\(x^3\)[/tex] term: [tex]\(2\)[/tex]
- [tex]\(x^2\)[/tex] term: [tex]\(0\)[/tex] (since there is no [tex]\(x^2\)[/tex] term, the coefficient is [tex]\(0\)[/tex])
- [tex]\(x\)[/tex] term: [tex]\(5\)[/tex]
- Constant term: [tex]\(3\)[/tex]
2. Add the coefficients of the corresponding terms:
- For [tex]\(x^3\)[/tex] terms: [tex]\(4 + 2 = 6\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(0 + 0 = 0\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-2 + 5 = 3\)[/tex]
- For the constant terms: [tex]\(-9 + 3 = -6\)[/tex]
3. Combine these results to form the polynomial:
The resulting polynomial after adding the coefficients is:
[tex]\(6x^3 + 0x^2 + 3x - 6\)[/tex]
Simplifying the polynomial expression (where [tex]\(0x^2\)[/tex] is typically omitted):
[tex]\(6x^3 + 3x - 6\)[/tex]
Therefore, the correct sum of the polynomials [tex]\(\left(4x^3 - 2x - 9\right) + \left(2x^3 + 5x + 3\right)\)[/tex] is:
[tex]\[6x^3 + 3x - 6\][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{6x^3 + 3x - 6} \][/tex]
