To simplify the given polynomial [tex]\(x^3 + 3x^2 - x + 2x^2 + 6x - 2\)[/tex], follow these steps:
1. Group the like terms together:
- [tex]\(x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(3x^2 + 2x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-x + 6x\)[/tex]
- Constant term: [tex]\(-2\)[/tex]
2. Combine the like terms:
- The [tex]\(x^3\)[/tex] term remains [tex]\(x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(3x^2 + 2x^2 = 5x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-x + 6x = 5x\)[/tex].
- The constant term remains [tex]\(-2\)[/tex].
3. Write out the simplified polynomial:
- [tex]\(x^3 + 5x^2 + 5x - 2\)[/tex]
Thus, the equivalent expression to the polynomial [tex]\(x^3 + 3x^2 - x + 2x^2 + 6x - 2\)[/tex] after full simplification is [tex]\(x^3 + 5x^2 + 5x - 2\)[/tex].
From the given options, this matches [tex]\(x^3 + 5x^2 + 5x - 2\)[/tex].
