To simplify the given polynomial [tex]\(x^3 + 3x^2 - x + 2x^2 + 6x - 2\)[/tex], we need to combine like terms. Let's follow the step-by-step simplification process:
1. Identify and combine the terms involving [tex]\(x^3\)[/tex]:
- The given polynomial has one term involving [tex]\(x^3\)[/tex]: [tex]\(x^3\)[/tex].
- Therefore, the coefficient of [tex]\(x^3\)[/tex] is 1.
2. Identify and combine the terms involving [tex]\(x^2\)[/tex]:
- The polynomial has two terms involving [tex]\(x^2\)[/tex]: [tex]\(3x^2\)[/tex] and [tex]\(2x^2\)[/tex].
- Combine these terms: [tex]\(3x^2 + 2x^2 = 5x^2\)[/tex].
- Therefore, the coefficient of [tex]\(x^2\)[/tex] is 5.
3. Identify and combine the terms involving [tex]\(x\)[/tex]:
- The polynomial has two terms involving [tex]\(x\)[/tex]: [tex]\(-x\)[/tex] and [tex]\(6x\)[/tex].
- Combine these terms: [tex]\(-x + 6x = 5x\)[/tex].
- Therefore, the coefficient of [tex]\(x\)[/tex] is 5.
4. Identify and combine the constant terms:
- There is one constant term in the polynomial: [tex]\(-2\)[/tex].
Combining all the simplified terms, we get:
[tex]\[ x^3 + 5x^2 + 5x - 2. \][/tex]
Thus, the expression that is equivalent to the given polynomial after fully simplifying it is:
[tex]\[ x^3 + 5x^2 + 5x - 2. \][/tex]
Therefore, the correct answer is:
[tex]\[ x^3 + 5x^2 + 5x - 2. \][/tex]
