The product of a binomial and a trinomial is [tex]\( x^3 + 3x^2 - x + 2x^2 + 6x - 2 \)[/tex]. Which expression is equivalent to this product after it has been fully simplified?

A. [tex]\( x^3 + 5x^2 + 5x - 2 \)[/tex]
B. [tex]\( x^3 + 2x^2 + 8x - 2 \)[/tex]
C. [tex]\( x^3 + 11x^2 - 2 \)[/tex]
D. [tex]\( x^3 + 10x^2 - 2 \)[/tex]



Answer :

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]